Reference Ray Research Articles

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.

Two-point paraxial traveltimes in an inhomogeneous anisotropic medium

SUMMARY We derive formulae for the approximate computation of two-point paraxial traveltimes (traveltimes between two points) for points arbitrarily chosen in a paraxial vicinity of a reference ray computed in a smoothly varying inhomogeneous anisotropic medium containing structural interfaces. The formulae have a form of the Taylor expansion in Cartesian coordinates of the two-point paraxial traveltime or its square to the quadratic terms. The coefficients of the expansion depend on quantities obtained by ray tracing in Cartesian coordinates and by dynamic ray tracing in ray-centred coordinates. Alternatively, the dynamic ray tracing can be performed in Cartesian coordinates. The advantages of the approach based on dynamic ray tracing in ray-centred coordinates are its efficiency and elimination of possible complications that may arise from the redundant fundamental solutions of dynamic ray tracing in Cartesian coordinates (the ray-tangent and non-eikonal solutions). As a by-product, we also obtain simple formulae for the slowness vectors at the two points in the paraxial vicinity of the reference ray. They belong to a paraxial ray passing through these points. Potential applications of the proposed formulae consist in the fast and flexible two-point traveltime calculations from sources to receivers specified in Cartesian coordinates and situated close to a reference ray, along which dynamic ray tracing has been performed. The formulae for the paraxial slowness vectors can be used in two-point ray tracing.

Focusing of a wave beam in an underwater sound channel

The use of a vertical radiating antenna array for generation of a wave beam propagating in an underwater sound channel along the reference ray trajectory is discussed. The method for selecting the starting field in the antenna aperture for maximum compression of a beam in the specified vicinity of the reference ray is proposed. The estimates showing up to what distances a beam can propagate while remaining narrow as compared to the range of depths between the rotation horizons have been obtained. The problem concerning the distances from the antenna array at which a beam can still be effectively focused in the vicinity of the selected reference ray point is investigated.

Perturbation expansions of complex-valued traveltime along real-valued reference rays

SUMMARY The eikonal equation in an attenuating medium has the form of a complex-valued Hamilton–Jacobi equation and must be solved in terms of the complex-valued traveltime (complex-valued action function). A very suitable approximate method for calculating the complex-valued traveltime right in real space is represented by the perturbation from the reference traveltime calculated along real-valued reference rays to the complex-valued traveltime defined by the complex-valued Hamilton–Jacobi equation. The real-valued reference rays are calculated using the reference Hamiltonian function. The perturbation Hamiltonian function is parametrized by one or more perturbation parameters, and smoothly connects the reference Hamiltonian function with the Hamiltonian function corresponding to a given complex-valued Hamilton–Jacobi equation. Both the reference Hamiltonian function and the perturbation Hamiltonian function may be constructed in different ways, yielding differently accurate perturbation expansions of traveltime. All present perturbation methods use reference rays calculated in a reference anisotropic non-attenuating medium. In this paper, the reference Hamiltonian function is constructed directly using the Hamiltonian function corresponding to a given complex-valued Hamilton–Jacobi equation, and the perturbation Hamiltonian function is linear with respect to the perturbation parameter. The direct construction of the reference Hamiltonian function from the given complex-valued Hamilton–Jacobi equation is very general and accurate, especially for homogeneous Hamiltonian functions of degree N=−1 with respect to the slowness vector.

Radiance Transfer Biclustering for Real-Time All-Frequency Biscale Rendering

We present a real-time algorithm to render all-frequency radiance transfer at both macroscale and mesoscale. At a mesoscale, the shading is computed on a per-pixel basis by integrating the product of the local incident radiance and a bidirectional texture function. While at a macroscale, the precomputed transfer matrix, which transfers the global incident radiance to the local incident radiance at each vertex, is losslessly compressed by a novel biclustering technique. The biclustering is directly applied on the radiance transfer represented in a pixel basis, on which the BTF is naturally defined. It exploits the coherence in the transfer matrix and a property of matrix element values to reduce both storage and runtime computation cost. Our new algorithm renders at real-time frame rates realistic materials and shadows under all-frequency direct environment lighting. Comparisons show that our algorithm is able to generate images that compare favorably with reference ray tracing results, and has obvious advantages over alternative methods in storage and preprocessing time.

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.

Formation of a narrow sound beam in the underwater sound channel by a vertical array

The possibility of using a vertical array for the generation of a narrow wave beam that propagates in the underwater sound channel along a given reference ray is discussed. The variational problem of choosing the initial field at the array aperture to provide the minimal possible average beam width along the propagation path of a fixed length is solved.

Perturbation Hamiltonians in heterogeneous anisotropic weakly dissipative media

SUMMARY Perturbation Hamiltonians simplify considerably the solution of various traveltime perturbation problems of seismic body waves propagating in heterogeneous, isotropic or anisotropic, weakly dissipative media. In addition to the phase-space coordinates, representing the Cartesian coordinates and Cartesian components of the slowness vector, the perturbation Hamiltonians also depend on one or more dimensionless perturbation parameters, which can be chosen in different ways. General traveltime perturbation procedures, based on the perturbation Hamiltonians, were proposed by Klimes. The evaluation of the perturbation expansions for the traveltime and for its derivatives does not require the computation of perturbed rays in these procedures. It consists in dynamic ray tracing along the reference ray, in the determination of the partial derivatives of the perturbation Hamiltonian with respect to the perturbation parameters, and in additional quadratures along a reference ray, computed in the reference medium. Different forms of the perturbation Hamiltonian yield different perturbation expansions with different accuracy and numerical efficiency of computations. In this paper, the special case of the perturbation Hamiltonian, herein referred to as the linear perturbation Hamiltonian, is used to compute the second-order perturbation expansion of the complex-valued traveltime and the first-order perturbation expansion of the complex-valued traveltime gradient. The advantage of the linear perturbation Hamiltonian is that it is linear with respect to perturbation parameters. Heterogeneous, perfectly elastic, isotropic or anisotropic, reference media are considered, in which reference rays can be calculated without problems by well-known methods. The derived perturbation expansions are valid very generally. For the isotropic reference model, the perturbed medium may be isotropic perfectly elastic, isotropic weakly dissipative, weakly anisotropic perfectly elastic or weakly anisotropic weakly dissipative. For the anisotropic reference model, the perturbed medium may be anisotropic perfectly elastic, or anisotropic weakly dissipative. Expressions for the attenuation vector, the dissipation filter, the global absorption factor, the direction-dependent scalar local quality factor and the attenuation coefficient are also derived for seismic body waves propagating in heterogeneous anisotropic weakly dissipative media.

Wave beam with minimum divergence in a smoothly inhomogeneous medium

We consider acoustic wave beams propagating in a smoothly inhomogeneous medium along a given reference ray. It is shown that in the aberration-free approximation, a Gaussian beam diverges with distance (on the average) less than any other beam with the same initial width. This result has been obtained by solving a variational problem that is similar to the well-known quantum-mechanical problem of seeking the quantum state with minimum uncertainty (coherent state). An example of the beam with minimum divergence in the realistic model of a deep-ocean acoustic waveguide is considered. An approximate analytical estimate for the amplitudes of normal modes forming the beam is obtained.

Numerical comparison of the isotropic-common-ray and anisotropic-common-ray approximations of the coupling ray theory

In the common-ray approximation of the coupling ray theory, the S-wave traveltimes in the coupling equations are approximated by the first-order perturbation expansion from the common reference ray. 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 traveltimes due to the perturbation from the common reference ray. The errors of S-wave traveltimes due to the common-ray approximation may be approximated by the second-order terms in the perturbation expansion. The first-order and second-order terms in the perturbation expansion can be calculated by simple numerical quadratures along the common reference ray. The common reference ray may be represented by the isotropic common ray or by the anisotropic common ray. The anisotropic-common-ray approximation of the coupling ray theory is more accurate than the isotropic-common-ray approximation. The equations for the first-order and second-order perturbation expansions of traveltime from both the isotropic and anisotropic common rays to the anisotropic-ray-theory rays have already been derived. We numerically test the equations in three smooth 1-D velocity models of differing degrees of anisotropy. We first compare the second-order perturbation expansions of traveltime from the isotropic common rays to the anisotropic common rays and vice versa. This comparison not only demonstrates the accuracy of the perturbation expansions, but also allows us to estimate the order of magnitude of the third-order, fourth-order and fifth-order terms in the perturbation expansions. The first-order and second-order perturbation expansions of traveltime from the isotropic common rays and from the anisotropic common rays to the anisotropic-ray-theory rays are then compared with the correct anisotropic-ray-theory traveltimes in order to illustrate the accuracy of the perturbation expansions and reliability of the error estimates. Finally, the synthetic seismograms obtained by the isotropic-common-ray and anisotropic-common-ray approximations of the coupling ray theory are compared with the more accurate coupling-ray-theory synthetic seismograms simulated by the second-order perturbation expansion of traveltime from the anisotropic common rays. In the numerical examples, the errors of the anisotropic-common-ray approximation of the coupling ray theory are considerably smaller than the errors of the isotropic-common-ray approximation.

Simulation of electron mirrors by the differential algebraic method

A differential algebraic (DA) method has been developed for the aberration analysis of electron mirrors, and a software package MIRROR_DA has been developed. Time is used instead of axial position as our independent variable, and a reference ray is also employed. A single DA ray is traced in the fields of the mirror system, performed on a non-standard extension of real number space called nDv, and all aberrations coefficients can be obtained simultaneously, in principle up to arbitrary order and with extremely high accuracy. To demonstrate our software, an example of tetrode mirror is presented. From the results by MIRROR_DA, it is shown that when adjusting the voltages of two middle electrodes, Cs and Cc change signs from positive to negative, and their values vary in a wide range: therefore this tetrode mirror can be used to cancel Cs and Cc of round lenses.

Simple and general method to calculate the dispersion properties of complex and aberrated stretchers-compressors

We propose a general method to calculate the dispersion of an arbitrary optical system. It is based on a nonlinear extension of the ABCD matrix model, where each optical element is described as an operator rather than a matrix. The deviation from a reference ray in terms of transverse position, angle, and phase as a function of wavelength is propagated through any optical system. This allows the calculation of all orders of dispersion, and also gives some insight in the space-time coupling phenomena such as spatial and angular chirp. This method is well-suited to compute the linear dispersive properties of complex and/or aberrated stretcher and compressor setups.

Numerical comparison of the isotropic-common-ray and anisotropic-common-ray approximations of the coupling ray theory

SUMMARY In the common-ray approximation of the coupling ray theory, the S-wave traveltimes in the coupling equations are approximated by the first-order perturbation expansion from the common reference ray. 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 traveltimes due to the perturbation from the common reference ray. The errors of S-wave traveltimes due to the common-ray approximation may be approximated by the second-order terms in the perturbation expansion. The first-order and second-order terms in the perturbation expansion can be calculated by simple numerical quadratures along the common reference ray. The common reference ray may be represented by the isotropic common ray or by the anisotropic common ray. The anisotropic-common-ray approximation of the coupling ray theory is more accurate than the isotropic-common-ray approximation. The equations for the first-order and second-order perturbation expansions of traveltime from both the isotropic and anisotropic common rays to the anisotropic-ray-theory rays have already been derived. We numerically test the equations in three smooth 1-D velocity models of differing degrees of anisotropy. We first compare the second-order perturbation expansions of traveltime from the isotropic common rays to the anisotropic common rays and vice versa. This comparison not only demonstrates the accuracy of the perturbation expansions, but also allows us to estimate the order of magnitude of the third-order, fourth-order and fifth-order terms in the perturbation expansions. The first-order and second-order perturbation expansions of traveltime from the isotropic common rays and from the anisotropic common rays to the anisotropic-ray-theory rays are then compared with the correct anisotropic-ray-theory traveltimes in order to illustrate the accuracy of the perturbation expansions and reliability of the error estimates. Finally, the synthetic seismograms obtained by the isotropic-common-ray and anisotropiccommon-ray approximations of the coupling ray theory are compared with the more accurate coupling-ray-theory synthetic seismograms simulated by the second-order perturbation expansion of traveltime from the anisotropic common rays. In the numerical examples, the errors of the anisotropic-common-ray approximation of the coupling ray theory are considerably smaller than the errors of the isotropic-common-ray approximation.

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.

Paraxial ray methods for anisotropic inhomogeneous media

ABSTRACTA 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.

Ray propagator matrices in three-dimensional anisotropic inhomogeneous layered media

SUMMARY Ray propagator matrices contain the complete solutions to the system of dynamic ray-tracing (DRT) equations connected with a given reference ray. They play an important role in studying the properties of complete four-parameteric systems of paraxial rays and offer many applications in both numerical modelling and practical interpretational problems of seismic ray fields in the high-frequency asymptotic approximation. Traditionally, ray propagator matrices have been expressed either in Cartesian or in ray-centred coordinates, connected with a reference ray. Both coordinate systems have certain advantages. For ray-centred coordinates, the dimensions of ray propagator matrices can be easily reduced from 6 × 6 to 4 × 4 (in a 3-D medium), thereby reflecting the strictly four-parameteric nature of a general paraxial ray field. On the other hand, in Cartesian coordinates, the computations are conceptually simpler and generally valid in isotropic and anisotropic media. In a Cartesian coordinate system, the DRT system and ray propagator matrices are well known both for isotropic and anisotropic layered media. In ray-centred coordinates, the DRT system and ray propagator matrices are known for isotropic layered structures and for anisotropic smooth media, but not for anisotropic media with structural interfaces. We propose a simple and invertible transformation between ray propagator matrices in both coordinate systems. It allows to perform conventional DRT in Cartesian coordinates, and to transform the resulting ray propagator matrix to ray-centred coordinates at any point of the ray where we need it. This avoids DRT in ray-centred coordinates altogether. Vice versa, we can compute, at any point of the reference ray, the ray propagator matrix in Cartesian coordinates by DRT in ray-centred coordinates. We propose several alternative versions of the transformation, each of them equally valid in isotropic and anisotropic media. For rays hitting a structural interface, the ray propagator matrix has to be transformed across the interface. The relevant transformation matrix is usually referred to as the interface propagator matrix. In Cartesian coordinates, the 6 × 6 interface propagator matrix has been published before, but in ray-centred coordinates only for the case of isotropic media. Based on the transformation from Cartesian to ray-centred coordinates, we present the 6 × 6 and 4 × 4 interface propagator matrices in ray-centred coordinates, valid for general isotropic and anisotropic media. The 4 × 4 interface propagator matrix in ray-centred coordinates can be factorized and used to derive the 4 × 4 surface-to-surface paraxial matrices. These matrices allow to relate the paraxial ray properties at different surfaces crossed by the reference ray and offer many important applications.

Design of dual Beam multi-wavelength UV-visible absorbance detectors based on CCD

Because the general multi-wavelength UV-Visible absorbance detector cannot avoid the noise and drift resulting from the intensity fluctuation of the light source, a dual beam multi-wavelength UV-Visible detector based on CCD was designed. The ray of light source is divided into a signal ray and a reference ray by the beam splitter after it passes through the chopper. The signal ray shines into the sample cell. The signal ray passing through the sample cell fails onto a concave mirror which focuses it onto a slot that is imaged on one portion of CCD by a concave grating. The reference ray is imaged on the other portion of CCD by the concave grating after the slot. The signal spectrum, the reference spectrum and the dark current of CCD can be measured on the same CCD under the cooperation of the optical system and accessorial circuits. The real-time compensation for the signal spectrum by using the reference spectrum and the dark current of CCD can effectively depress the noise and drift of the detector. The short-term noise is 10−5 AU and the drift is 10−4 AU/h.

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.

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.

Caustics and weakly divergent beams in oceanic waveguides

It is found that the weakly divergent beam formed around the reference ray with minimal cycle length does not stand out in intensity against the background of the total field. It is shown that, under the same conditions, another beam stands out in intensity: it is the caustic beam composed of the system of caustic lines, which are the envelopes of the family of rays forming the weakly divergent beam.