Relation of the Wave-Propagation Metric Tensor to the Curvatures of the Slowness and Ray-Velocity Surfaces

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.     

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.     

Paraxial ray methods for anisotropic inhomogeneous media

A new formalism of surface-to-surface paraxial matrices allows a very general and flexible formulation of the paraxial ray theory, equally valid in anisotropic and isotropic inhomogeneous layered media. The formalism is based on conventional dynamic ray tracing in Cartesian coordinates along a reference ray. At any user-selected pair of points of the reference ray, a pair of surfaces may be defined. These surfaces may be arbitrarily curved and oriented, and may represent structural interfaces, data recording surfaces, or merely formal surfaces. A newly obtained factorization of the interface propagator matrix allows to transform the conventional 6 × 6 propagator matrix in Cartesian coordinates into a 6 × 6 surface-to-surface paraxial matrix. This matrix defines the transformation of paraxial ray quantities from one surface to another. The redundant non-eikonal and ray-tangent solutions of the dynamic ray-tracing system in Cartesian coordinates can be easily eliminated from the 6 × 6 surface-to-surface paraxial matrix, and it can be reduced to 4 × 4 form. Both the 6 × 6 and 4 × 4 surface-to-surface paraxial matrices satisfy useful properties, particularly the symplecticity. In their 4 × 4 reduced form, they can be used to solve important boundary-value problems of a four-parametric system of paraxial rays, connecting the two surfaces, similarly as the well-known surface-to-surface matrices in isotropic media in ray-centred coordinates. Applications of such boundary-value problems include the two-point eikonal, relative geometrical spreading, Fresnel zones, the design of migration operators, and more.     

Errors Due to the Common Ray Approximations of the Coupling Ray Theory

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.     

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.     

A Fast Sweeping Scheme for Calculating P Wave First-Arrival Travel Times in Transversely Isotropic Media with an Irregular Surface

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.     

Errors due to the anisotropic-common-ray approximation of the coupling ray theory

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.     

Kinematic hypocentre determination using the paraxial ray approximation

Summary The robust nonlinear approach by Tarantola and Valette, consisting in direct evaluation of the "probability" density function, is supplemented with the paraxial ray approximation of the travel time. A sufficiently dense 2-parametric system of rays from each receiver is evaluated only once for all hypocentre determinations. The interpolation formulae for the travel times apply to all travel-time branches. Their derivation is based on the summation of Gaussian packets. The proposed algorithm for determining the hypocentre is able to find all of its possible locations.     

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.     

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.     

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.     

Interpretation of seismic reflection records: Direct calculation of interval velocities and layer thicknesses from travel times

In reflection surveys and velocity analysis, calculations of interval velocities and layer-thicknesses of a multilayered horizontal structure are often based on Dix's equation which requires the travel times at zero offsets and a prior estimate of the root mean squared velocities.In this paper a method is presented which requires only the reflection travel-time data. A set of equations are derived which relate the interval velocity and thickness of a layer to the reflection travel time from the top and the bottom of that layer, the offset distances and the ray parameter. It is shown that the difference of the offset distances and the difference of the picked travel times of any reflected rays with the same value of ray parameter from the top and the bottom of a horizontal layer can be used to calculate the interval velocity and thickness of that layer.     

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.     

True Amplitude Migration Weights from Travel Times

— 3-D amplitude preserving prestack migration of the Kirchhoff type is a task of high computational effort. A substantial part of this effort is spent on the calculation of proper weight functions for the diffraction stack. We propose a new strategy to compute the migration weights directly from coarse gridded travel-time data which are in any event needed for the summation along diffraction time surfaces. The technique employs second-order travel-time derivatives that contain all necessary information on the weight functions. Their determination alone from travel times significantly reduces the requirements in computational time and particularly storage, since it is done on the fly. Application of the method shows good accordance between numerical and analytical results for the simple types of models considered in this study.     

Transformation of spatial and perturbation derivatives of travel time at a curved interface between two arbitrary media

We consider the partial derivatives of travel time with respect to both spatial coordinates and perturbation parameters. These derivatives are very important in studying wave propagation and have already found various applications in smooth media without interfaces. In order to extend the applications to media composed of layers and blocks, we derive the explicit equations for transforming these travel–time derivatives of arbitrary orders at a general smooth curved interface between two arbitrary media. The equations are applicable to both real–valued and complex–valued travel time. The equations are expressed in terms of a general Hamiltonian function and are applicable to the transformation of travel–time derivatives in both isotropic and anisotropic media. The interface is specified by an implicit equation. No local coordinates are needed for the transformation.     

Joint analysis of seismological data by the USSO and DSS stations in the Caucasus region

This paper deals with a procedure of a joint analysis of seismic data from earthquakes and those obtained by DSS. The DSS data are used as a first approximation to construct a two-dimensional model of the medium made up of individual blocks. These models serve as a basis when constructing specific three-dimensional travel-time curves. These travel-time curves are further used for the calculation of hypocenter parameters in a laterally inhomogeneous block medium.The hypocenter field and the travel times obtained are input data for the computation of three-dimensional fields of velocities in earthquake focal zones. Results of applying the proposed procedure to the Caucasus region are presented.     

A new approximation of the velocity-depth distribution and its application to the computation of seismic wave fields

Summary A new approximation of the velocity-depth distribution in a vertically inhomogeneous medium is suggested. This approximation guarantees the continuity of velocity and of its first and second derivatives and does not generate false low-velocity zones. It is very suitable for the computations of seismic wave fields in vertically inhomogeneous media by ray methods and its modifications, as it removes many false anomalies from the travel-time and amplitude-distance curves of seismic body waves. The ray integrals can be evaluated in a closed form; the resulting formulae for rays, travel times and geometrical spreading are very simple. They do not contain any transcendental functions (such asln (x) orsin ^–1, (x)) like other approximations; only the evaluation of one square root and of certain simple arithmetic expressions for each layer is required. From a computational point of view, the evaluation of ray integrals and of geometrical spreading is only slightly slower than for a system of homogeneous parallel layers and even faster than for a piece-wise linear approximation.     

Three-dimensional Travel-time Calculation Based on Fermat's Principle

— We present a new travel-time calculation method based on Fermat's principle. In the method, travel times are recursively calculated on horizontal planes of increasing depth. For typical configurations of exploration geophysics, the distance between the planes is on the order of 100 m. The travel times on the first plane are calculated by connecting straight ray segments from the source point to the grid points on the plane and integrating the slowness along each segment. The times on the other planes are calculated by finding the minimum of the combination of the times on the plane above plus the additional time along segments connecting grid points on the two planes. The travel-time calculation method is designed for calculating either the first arrival times, or the time of the shortest travel path arrival. The method is extended to handle vertically transverse isotropic (VTI) media by an approach which increases the computing time only slightly. The algorithm is tested against synthetic examples for isotropic and VTI wave propagation.     

Prediction uncertainty for tracer migration in random heterogeneities with multifractal character

Travel-time statistics for non-reacting tracers in fractal and multifractal media are addressed through numerical simulations. The logarithm of hydraulic conductivity is modeled using fractional Brownian motion (fBm) and more recently developed multifractal model based on bounded fractional Levy motion (bfLm). These models have been shown previously to accurately reproduce statistical properties of large conductivity datasets. The ensemble-mean travel time increases nearly linearly with travel distance and the variance in the travel time increases nearly parabolically with travel distance. This is consistent with near-field analytical approximations developed for non-fractal media and suggests that these analytical results may have some degree of robustness to non-ideal features in the random-field models. The magnitudes of the travel-time moments are dependent on the system size. For fBm media, this size dependence can be explained using an effective variance that increases with increasing size of the flow system. However, the magnitudes of the travel-time moments are also sensitive to other non-ideal effects such as deviations from Gaussian behavior. This sensitivity illustrates the need for careful aquifer characterization and conditional numerical simulation in practical situations requiring accurate estimates of uncertainty in the plume position.     

Inverse kinematic problem for a random gradient medium in geometric optics approximation

Scattering at random inhomogeneities in a gradient medium results in systematic deviations of the rays and travel times of refracted body waves from those corresponding to the deterministic velocity component. The character of the difference depends on the parameters of the deterministic and random velocity component. However, at great distances to the source, independently of the velocity parameters (weakly or strongly inhomogeneous medium), the most probable depth of the ray turning point is smaller than that corresponding to the deterministic velocity component, the most probable travel times also being lower. The relative uncertainty in the deterministic velocity component, derived from the mean travel times using methods developed for laterally homogeneous media (for instance, the Herglotz-Wiechert method), is systematic in character, but does not exceed the contrast of velocity inhomogeneities by magnitude. The gradient of the deterministic velocity component has a significant effect on the travel-time fluctuations. The variance at great distances to the source is mainly controlled by shallow inhomogeneities. The travel-time flucutations are studied only for weakly inhomogeneous media.