A note on two-point paraxial travel times

Recently, several expressions for the two-point paraxial travel time in laterally varying, isotropic or anisotropic layered media were derived. The two-point paraxial travel time gives the travel time from point S′ to point R′, both these points being situated close to a known reference ray Ω, along which the ray-propagator matrix was calculated by dynamic ray tracing. The reference ray and the position of points S′ and R′ are specified in Cartesian coordinates. Two such expressions for the two-point paraxial travel time play an important role. The first is based on the 4 × 4 ray propagator matrix, computed by dynamic ray tracing along the reference ray in ray-centred coordinates. The second requires the knowledge of the 6 × 6 ray propagator matrix computed by dynamic ray tracing along the reference ray in Cartesian coordinates. Both expressions were derived fully independently, using different methods, and are expressed in quite different forms. In this paper we prove that the two expressions are fully equivalent and can be transformed into each other.     

Review of the Anisotropic Interface Ray Propagator: Symplecticity,Eigenvalues, Invariants and Applications

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.     

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.     

Transformation relations for second-order derivatives of travel time in anisotropic media

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.     

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.     

PARABOLIC AND HYPERBOLIC PARAXIAL TWO-POINT TRAVELTIMES IN 3D MEDIA1

The 4 × 4 T -propagator matrix of a 3D central ray determines, among other important seismic quantities, second-order (parabolic or hyperbolic) two-point traveltime approximations of certain paraxial rays in the vicinity of the known central ray through a 3D medium consisting of inhomogeneous isotropic velocity layers. These rays result from perturbing the start and endpoints of the central ray on smoothly curved anterior and posterior surfaces. The perturbation of each ray endpoint is described only by a two-component vector. Here, we provide parabolic and hyperbolic paraxial two-point traveltime approximations using the T -propagator to feature a number of useful 3D seismic models, putting particular emphasis on expressing the traveltimes for paraxial primary reflected rays in terms of hyperbolic approximations. These are of use in solving several forward and inverse seismic problems. Our results simplify those in which the perturbation of the ray endpoints upon a curved interface is described by a three-component vector. In order to emphasize the importance of the hyperbolic expression, we show that the hyperbolic paraxial-ray traveltime (in terms of four independent variables) is exact for the case of a primary ray reflected from a planar dipping interface below a homogeneous velocity medium.     

Thermoelastic deformation of a transversely isotropic and layered half-space by surface loads and internal sources

A general method is developed for the study of transient thermoelastic deformation in a transversely isotropic and layered half-space by surface loads and internal sources. A Laplace transform is first applied to the field quantities; Cartesian and cylindrical systems of vector functions are then introduced for reducing the basic equations to three sets of simultaneous linear differential equations. General solutions are obtained from these sets, and propagator matrices from the solutions by a partitioned matrix method.

Source functions for a variety of sources are derived in the Cartesian and cylindrical systems, and the Laplace transformed expressions of the field variables at the surface presented explicitly in the two systems in terms of a layer matrix. The effect of gravity is included by multiplying simply an effect matrix resulting from the modification of continuity conditions at the surface and the layer interfaces.

It should be noted that the present analytical method has great advantages over either the classical thin plate approach or the finite element method, and that the present result can be reduced directly to the solutions of the corresponding isotropic case.     

Phase shift of the Green tensor due to caustics in anisotropic media

Equations are presented to determine the phase shift of the amplitude of the elastic Green tensor due to both simple (line) and point caustics in anisotropic media. The phase-shift rules for the Green tensor are expressed in terms of the paraxial-ray matrices calculated by dynamic ray tracing. The phase-shift rules are derived both for 2×2 paraxial-ray matrices in ray-centred coordinates and for 3×3 paraxial-ray matrices in general coordinates. The reciprocity of the phase shift of the Green tensor is demonstrated. Then a simple example is given to illustrate the positive and negative phase shifts in anisotropic media, and also to illustrate the reciprocity of the phase shift of the Green tensor.     

Estimation of geological dip and curvature from time‐migrated zero‐offset reflections in heterogeneous anisotropic media

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.     

Estimation of geological dip and reflector curvature from zero-offset seismic reflections in heterogeneous anisotropic media

Depth conversion of selected seismic reflections is a valuable procedure to position key reflectors in depth in a process of constructing or refining a depth-velocity model. The most widespread example of such procedure is the so-called map migration, in which normal-incidence, zero-offset (stacked) seismic data are employed. Since the late seventies and early eighties, under the assumption of an isotropic velocity model, map migration algorithms have been devised to convert traveltime and its first and second derivatives into reflector position, dip and curvatures in depth. In this work we revisit map migration to improve the existing algorithms in the following accounts: (a) We allow for fully anisotropic media; (b) In contrast to simple planar measurement surface, arbitrary topography is allowed, thus enlarging the algorithms applicability and (c) Derivations and results are much simplified upon the use of the methodology of surface-to-surface paraxial matrices.     

Eigenrays in 3D heterogeneous anisotropic media,Part II: Dynamics

This paper is the second in a sequel of two papers and dedicated to the computation of paraxial rays and dynamic characteristics along the stationary rays obtained in the first paper. We start by formulating the linear, second‐order, Jacobi dynamic ray tracing equation. We then apply a similar finite‐element solver, as used for the kinematic ray tracing, to compute the dynamic characteristics between the source and any point along the ray. The dynamic characteristics in our study include the relative geometric spreading and the phase correction due to caustics (i.e. the amplitude and the phase of the asymptotic form of the Green's function for waves propagating in 3D heterogeneous general anisotropic elastic media). The basic solution of the Jacobi equation is a shift vector of a paraxial ray in the plane normal to the ray direction at each point along the central ray. A general paraxial ray is defined by a linear combination of up to four basic vector solutions, each corresponds to specific initial conditions related to the ray coordinates at the source. We define the four basic solutions with two pairs of initial condition sets: point–source and plane‐wave. For the proposed point–source ray coordinates and initial conditions, we derive the ray Jacobian and relate it to the relative geometric spreading for general anisotropy. Finally, we introduce a new dynamic parameter, similar to the endpoint complexity factor, presented in the first paper, used to define the measure of complexity of the propagated wave/ray phenomena. The new weighted propagation complexity accounts for the normalized relative geometric spreading not only at the receiver point, but along the whole stationary ray path. We propose a criterion based on this parameter as a qualifying factor associated with the given ray solution. To demonstrate the implementation of the proposed method, we use several isotropic and anisotropic benchmark models. For all the examples, we first compute the stationary ray paths, and then compute the geometric spreading and analyse these trajectories for possible caustics. Our primary aim is to emphasize the advantages, transparency and simplicity of the proposed approach.     

Paraxial ray theory in general coordinates

Summary Principles of an alternative approach to the ray theory and paraxial ray approximation (PRA) theory are discussed. Invariant equations for the ray, the eikonal equation and basic equations of the PRA-theory are derived on the basis of Riemannian geometry. Paraxial rays and paraxial time field equations in general curvilinear coordinates are shown as an example of application.     

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     

Time‐harmonic response of non‐homogeneous elastic unbounded domains using the scaled boundary finite‐element method

The scaled boundary finite‐element method is extended to simulate time‐harmonic responses of non‐homogeneous unbounded domains with the elasticity modulus and mass density varying as power functions of spatial coordinates. The unbounded domains and the elasticity matrices are transformed to the scaled boundary coordinates. The scaled boundary finite‐element equation in displacement amplitudes are derived directly from the governing equations of elastodynamics. To enforce the radiation condition at infinity, an asymptotic expansion of the dynamic‐stiffness matrix for high frequency is developed. The dynamic‐stiffness matrix at lower frequency is obtained by numerical integration of ordinary differential equations. Only the boundary is discretized yielding a reduction of the spatial dimension by one. No fundamental solution is required. Material anisotropy is modelled without additional efforts. Examples of two‐ and three‐dimensional non‐homogeneous isotropic and transversely isotropic unbounded domains are presented. The results demonstrate the accuracy and simplicity of the scaled boundary finite‐element method. Copyright © 2005 John Wiley & Sons, Ltd.     

Ray tracing for continuously rotated local coordinates belonging to a specified anisotropy

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.     

Application of a second-order paraxial boundary condition to problems of dynamics of circular foundations on a porous layered half-space

A half-space finite element and a consistent transmitting boundary in a cylindrical coordinate system are developed for analysis of rigid circular (or cylindrical) foundations in a water-saturated porous layered half-space. By means of second-order paraxial approximations of the exact dynamic stiffness for a half-space in plane-strain and antiplane-shear conditions, the corresponding approximation for general three-dimensional wave motion in a Cartesian coordinate system is obtained and transformed in terms of cylindrical coordinates. Using the paraxial approximations, the half-space finite element and consistent transmitting boundary are formulated in a cylindrical coordinate system. The development is verified by comparison of dynamic compliances of rigid circular foundations with available published results. Examination of the advantage of the paraxial condition vis-á-vis the fixed condition shows that the former achieves substantial gain in computational effort. The developed half-space finite element and transmitting boundary can be employed for accurate and effective analysis of foundation dynamics and soil–structure interaction in a porous layered half-space.     

Spatial derivatives and perturbation derivatives of amplitude in isotropic and anisotropic media

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.     

Propagator matrix formulation of heat transfer in spherically stratified media

Summary In the paper we have transformed the steady and unsteady conductive heat transfer differential equation in spherical coordinates into a system of first order differential equations and processed them by method of propagator matrices to extrapolate the known surface heat flux and temperature to any desired depth. The elements of propagator matrices have been summarised for various piecewise continuous conductivity and rate of heat generation functions to approximate inhomogeneities in the earth. In the analysis the rate of heat generation is either assumed to depend linearly upon temperature or correspond to first order irreversible chemical reactions.     

Ray-centred coordinate systems in anisotropic media

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.     

层状介质中地震面波频散函数和体波广义反射系数的计算

本文在B、P、C坐标中给出了弹性波在横向均匀介质中的传播矩阵,并将其表示为五个形式简单的矩阵乘积,其中有四个矩阵是与频率无关的。我们用传播矩阵的分离方式,将Abo-Zena（1979）算法大量简化。用本文的算法计算综合地震图时,计算量要比文献[2,7,8]的算法少一半以上。 文中还给出了地震面波频散函数和体波广义反射系数的快速计算步骤。