Ray Tracing Equations Research Articles

First-orderP-wave ray synthetic seismograms in inhomogeneous weakly anisotropic media

SUMMARY We propose approximate equations for P-wave ray theory Green’s function for smooth inhomogeneous weakly anisotropic media. Equations are based on perturbation theory, in which deviations of anisotropy from isotropy are considered to be the first-order quantities. For evaluation of the approximate Green’s function, earlier derived first-order ray tracing equations and in this paper derived first-order dynamic ray tracing equations are used. The first-order ray theory P-wave Green’s function for inhomogeneous, weakly anisotropic media of arbitrary symmetry depends, at most, on 15 weak-anisotropy parameters. For anisotropic media of higher-symmetry than monoclinic, all equations involved differ only slightly from the corresponding equations for isotropic media. For vanishing anisotropy, the equations reduce to equations for computation of standard ray theory Green’s function for isotropic media. These properties make the proposed approximate Green’s function an easy and natural substitute of traditional Green’s function for isotropic media. Numerical tests for configuration and models used in seismic prospecting indicate negligible dependence of accuracy of the approximate Green’s function on inhomogeneity of the medium. Accuracy depends more strongly on strength of anisotropy in general and on angular variation of phase velocity due to anisotropy in particular. For example, for anisotropy of about 8 per cent, considered in the examples presented, the relative errors of the geometrical spreading are usually under 1 per cent; for anisotropy of about 20 per cent, however, they may locally reach as much as 20 per cent.

Approximate ray tracing for qP-waves in inhomogeneous layered media with weak structural anisotropy

We present approximate equations for qP-wave ray tracing and paraxial ray tracing in inhomogeneous layered weakly transversely isotropic (TI) media. Inside layers, the symmetry axis direction of the TI medium is allowed to vary continuously. Approximate equations are based on perturbation theory in which deviations of anisotropy from isotropy are considered to be first-order quantities. For qP-waves propagating in a TI medium, the approximate ray-tracing and paraxial ray-tracing equations depend on three parameters and two angles defining the direction of the symmetry axis. We also present the boundary conditions at interfaces for first-order rays and paraxial rays and compute reflection/transmission coefficients to the first order. All the quantities required for evaluation of the Green’s function are calculated to the first order, except the traveltime that is calculated to the second order. The accuracy of the presented algorithm is verified on simple models with respect to exact ray-tracing results. For anisotropy of about [Formula: see text], considered in the examples presented, the relative errors in traveltime and amplitude are less than [Formula: see text] and [Formula: see text], respectively. We then consider media where the symmetry axis is conformable with the structure, named structural transverse isotropy (STI). The method is applied to simple STI models and to a realistic overthrust model, showing significant differences with vertical transverse isotropy (VTI) models not only for traveltimes, but also for amplitudes.

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.

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.

Modeling acoustic propagation of airgun array pulses recorded on tagged sperm whales (Physeter macrocephalus)

In 2002 and 2003, tagged sperm whales (Physeter macrocephalus) were experimentally exposed to airgun pulses in the Gulf of Mexico, with the tags providing acoustic recordings at measured ranges and depths. Ray trace and parabolic equation (PE) models provided information about sound propagation paths and accurately predicted time of arrival differences between multipath arrivals. With adequate environmental information, a broadband acoustic PE model predicted the relative levels of multipath arrivals recorded on the tagged whales. However, lack of array source signature data limited modeling of absolute received levels. Airguns produce energy primarily below 250 Hz, with spectrum levels about 20-40 dB lower at 1 kHz. Some arrivals recorded near the surface in 2002 had energy predominantly above 500 Hz; a surface duct in the 2002 sound speed profile helps explain this effect, and the beampattern of the source array also indicates an increased proportion of high-frequency sound at near-horizontal launch angles. These findings indicate that airguns sometimes expose animals to measurable sound energy above 250 Hz, and demonstrate the influences of source and environmental parameters on characteristics of received airgun pulses. The study also illustrates that on-axis source levels and simple geometric spreading inadequately describe airgun pulse propagation and the extent of exposure zones.

Self‐consistent model of magnetospheric ring current and propagating electromagnetic ion cyclotron waves: Waves in multi‐ion magnetosphere

The further development of a self‐consistent theoretical model of interacting ring current ions and electromagnetic ion cyclotron waves (Khazanov et al., 2003) is presented. In order to adequately take into account wave propagation and refraction in a multi‐ion magnetosphere, we explicitly include the ray tracing equations in our previous self‐consistent model and use the general form of the wave kinetic equation. This is a major new feature of the present model and, to the best of our knowledge, the ray tracing equations for the first time are explicitly employed on a global magnetospheric scale in order to self‐consistently simulate the spatial, temporal, and spectral evolution of the ring current and of electromagnetic ion cyclotron waves. To demonstrate the effects of EMIC wave propagation and refraction on the wave energy distribution and evolution, we simulate the May 1998 storm. The main findings of our simulation can be summarized as follows. First, owing to the density gradient at the plasmapause, the net wave refraction is suppressed, and He+‐mode grows preferably at the plasmapause. This result is in total agreement with previous ray tracing studies and is very clearly found in presented B field spectrograms. Second, comparison of global wave distributions with the results from another ring current model (Kozyra et al., 1997) reveals that this new model provides more intense and more highly plasmapause‐organized wave distributions during the May 1998 storm period. Finally, it is found that He+‐mode energy distributions are not Gaussian distributions and most important that wave energy can occupy not only the region of generation, i.e., the region of small wave normal angles, but all wave normal angles, including those to near 90°. The latter is extremely crucial for energy transfer to thermal plasmaspheric electrons by resonant Landau damping and subsequent downward heat transport and excitation of stable auroral red arcs.

Ray trace calculation of ionospheric propagation at lower frequencies

The Raytrace/Ionospheric Conductivity and Electron Density–Bent‐Gallagher model has been revised to make it applicable to ionospheric propagation at low radio frequencies (0.5–5.0 MHz), where the ionosphere and magnetic anisotropy drastically alter propagation paths and provide a severe test of propagation model algorithms. The necessary revisions are discussed, and the model is applied to the problem of ionospheric penetration from a source below the ionosphere to a receiver above the ionosphere. It is necessary to include the electron collision frequency in the Appleton‐Hartree index of refraction in order to permit ionospheric penetration for radio frequencies below the maximum plasma frequency (e.g., whistler modes). The associated reformulation of the ray trace equations for a complex index of refraction is straightforward. Difficulties with numerical methods are cited for the lowest frequencies, and future improvements are indicated.

Modeling the Radiative Properties of Microscale Random Roughness Surfaces

The radiative properties of engineering surfaces with microscale surface texture or topography (patterned or random roughness and coating or multi-layer) are of fundamental and practical importance. In the rapid thermal processing or arc/flash-assisted heating of silicon wafers, the control of thermal energy deposition through radiation and the surface temperature measurement using optical pyrometry require in-depth knowledge of the surface radiative properties. These properties are temperature, wavelength, doping level, and surface topography dependent. It is important that these properties can be modeled and predicted with high accuracy to meet very stringent temperature control and monitor requirements. This study solves the Maxwell equations that describe the electromagnetic wave reflection from the one-dimensional random roughness surfaces. The surface height conforms to the normal distribution, i.e., a Gaussian probability density function distribution. The numerical algorithm of Maxwell equations’ solution is based on the well-developed finite difference time domain (FDTD) scheme and near-to-far-field transformation. Various computational modeling issues that affect the accuracy of the predicted properties are quantified and discussed. The model produces the bi-directional reflectivity and is in good agreement with the ray tracing and integral equation solutions. The predicted properties of a perfectly electric conductor and silicon surfaces are compared and discussed.

Matrix-based paraxial skew ray-tracing in 3D systems with non-coplanar optical axis

Matrix-based Gaussian paraxial ray-tracing is commonly used for first estimates in early design stages of optical systems. However, the usual techniques are suitable only for rays in systems with a common straight-line optical axis, and cannot be used in systems containing mirrors and/or prisms. Attard extended these methods to matrix-based skew ray-tracing for systems containing cylindrical lenses with orthogonal cylinder axes and spherical lenses with combinations of cylindrical lenses, but had problems with non-straight and non-coplanar optical relations. This paper develops a novel general matrix method for paraxial skew ray-tracing in systems with non-coplanar optical axes containing spherical and flat boundary surfaces. First-order Taylor series expansion is used to approximate skew ray-tracing equations in simple repetitive linear matrix form. Sufficiently good accuracy is obtained if the proposed method is restricted to skew rays in the immediate neighborhood of the optical axis, as demonstrated by numerical examples. This study extends earlier Matrix-based paraxial ray-tracing design technique to include non-coplanar systems containing mirrors and/or reflecting prisms such as projectors, with special application potential for first estimates in early design stages of 3D optical systems.

Aberrations of self-imaged patterns with two-dimensional periodicity

We develop a geometrical theory of aberration in the self-imaged patterns with two-dimensional periodicity. The patterns are considered to be a crossed-line grating or a periodic array of finite apertures. We first derive the raytracing equations for determining the optical path of a self-imaging ray. We then find the third- and fifth-order contributions to the wavefront aberration which arise from the difference between the optical paths of a self-imaging ray and of an actual ray. We also derive the expression of the ray aberration from the wavefront aberration. The ray aberration is classified into five distinct types by analogy with the cases in a refracting lens system. We show that the overall ray aberration is entirely undercorrected and the aberrated image patch is decentered from an ideal image point in the direction parallel to the direction tangent vector of a chief ray. The image evaluation technique discussed here will be useful in various applications related to self-image formation of two-dimensionally periodic patterns.

First-order ray tracing for qP waves in inhomogeneous, weakly anisotropic media

We propose approximate ray-tracing equations for qP-waves propagating in smooth, inhomogeneous, weakly anisotropic media. For their derivation, we use perturbation theory, in which deviations of anisotropy from isotropy are considered to be the first-order quantities. The proposed ray-tracing equations and corresponding traveltimes are of the first order. Accuracy of the traveltimes can be increased by calculating a secondorder correction along first-order rays.The first-order ray-tracing equations for qP-waves propagating in a general weakly anisotropic medium depend on only 15 weak-anisotropy parameters (generalization of Thomsen’s parameters). The equations are thus considerably simpler than the exact ray-tracing equations. For higher-symmetry anisotropic media the equations differ only slightly from equations for isotropic media. They can thus substitute for the traditional isotropic ray tracers used in seismic processing. For vanishing anisotropy, the first-order ray-tracing equations reduce to standard, exact ray-tracing equations for isotropic media. Numerical tests for configuration and models used in seismic prospecting indicate negligible dependence of accuracy of calculated traveltimes on inhomogeneity of the medium. For anisotropy of about 8%, considered in the examples presented, the relative errors of the traveltimes, including the second-order correction, are well under 0.05%; for anisotropy of about 20%, they do not exceed 0.3%.

Ray-tracing and parabolic equation methods in the modeling of a tropospheric microwave link

An experimental tropospheric propagation system was operated for several years on a 50-km link in Ontario. The system provided information on the amplitude and angle-of-arrival of individual signal components under anomalous propagation conditions. Modeling of the observed activity can be attempted using various techniques and the use of ray-tracing and parabolic equation approaches are considered here in the light of the experimental measurements. Emphasis is placed on the latter which is capable of accurately including the effects of the ground in the model.

Analytic solution of ray-tracing equations for a linearly inhomogeneous and elliptically anisotropic velocity model

We study wave propagation in anisotropic inhomogeneous media. Specifically, we formulate and analytically solve the ray-tracing equations for the factorized model with wavefront velocity increasing linearly with depth and depending elliptically on direction. We obtain explicit expressions for traveltime, wavefront (phase) angle, and ray (group) velocity and angle, and study these seismological quantities for a model that successfully describes field measurements in the Western Canada Basin. By considering numerical examples, we also show that the difference between the wavefront and ray velocities depends only slightly on the anisotropy parameter, whereas the difference between the wavefront and ray angles is, in a first-order approximation, linear in the anisotropy parameter.

Thermospheric responses to gravity waves: Influences of increasing viscosity and thermal diffusivity

A gravity wave anelastic dispersion relation is derived that includes molecular viscosity and thermal diffusivity to explore the damping of high‐frequency gravity waves in the thermosphere. The time dependence of the wave amplitudes and general ray trace equations are also derived. In the special case that the thermal structure is isothermal and the Prandtl number (Pr) equals 1, exact linear solutions are obtained. For high‐frequency gravity waves with ωIr/N ≪ 1 an upward propagating gravity wave dissipates at an altitude given by ≃ z1 + H ln(ωIr/2H∣m∣3ν1), where H is the density scale height, N is the buoyancy frequency, ν1 is the viscosity at z = z1, and ωIr and m are the gravity wave intrinsic frequency and vertical wave number, respectively. Thus high‐frequency gravity waves with large vertical wavelengths dissipate at the highest altitudes, resulting in momentum and energy inputs extending to very high altitudes. We find that the vertical wavelength of a gravity wave with an initially large vertical wavelength decreases significantly by the time it dissipates just below where it begins to reflect. The effect of diffusion on a gravity wave is similar to the effect of shear in the sense that as the molecular viscosity and thermal diffusivity increase due to decreasing background density, the intrinsic frequency plus mν/H decreases and the vertical wave number increases in order to satisfy the dispersion relation for Pr = 1. We also briefly explore the results with different Prandtl numbers using numerical ray tracing. Gravity waves in a Pr = 0.7 environment dissipate just a few kilometers below those in a Pr = 1 environment when H = 7 km, showing the utility of the analytic, Pr = 1 solutions.

First-order ray tracing forqSwaves in inhomogeneous weakly anisotropic media

SUMMARY First-order ray tracing equations and paraxial ray tracing system are derived for quasi-shear waves propagating in a smooth inhomogeneous weakly anisotropic medium. The first-order equations are based on explicit formulae in terms of slowness vector and elastic parameters. They only depend on 15 independent combinations of elastic parameters. For isotropic media, they reduce to exact equations. The behaviour of the approximate equations is studied in the vicinity of singularities. Ray tracing equations behave properly if the wave mode is changed when the ray crosses the conical or the intersection singularity, but the paraxial ray tracing system may be singular. A second-order correction of the traveltime is obtained by integration along the approximate rays. Comparison of the exact and approximate traveltimes in transversely isotropic and orthorhombic homogeneous media shows small relative errors except in the close vicinity of cusps.

Paraxial theory for birefringent lenses

The generalized ray tracing for the extraordinary ray through uniaxial crystals developed by Avendaño-Alejo and Stavroudis [J. Opt. Soc. Am. A 19, 1674 (2002)] has been applied to derive paraxial refracting equations. Paraxial equations are derived for three cases where the incident, ordinary, and extraordinary rays lie in the incident plane: (a) the crystal axis is parallel to the optical axis, (b) the crystal axis is orthogonal to the optical axis and lies in the plane of incidence, and (c) the crystal axis is orthogonal to both the optical axis and the incident plane. The paraxial ray-tracing equations for the extraordinary ray are represented by matrix operators. The elements of the matrix system give all the information of the focal points and of the principal points. Gaussian formulas are derived, and some examples are presented.

Geometrical aberrations of self-imaged line gratings

We analyze the properties of a self-imaging system from the point of view of aberration theory. We examine analytically and numerically the geometrical aberrations that are observed in the self-image of a parallel-line grating. We first derive the raytracing equations for determining the optical path of a self-imaging ray with the order of diffraction l . We then obtain the third- and fifth-order contributions to ray aberrations which arise from the difference between the optical paths of a self-imaging ray and of an actual ray. We show that the overall ray aberrations are entirely undercorrected. The ray aberrations approach zero as the ratio of the grating constant to the wavelength of light becomes large enough. In a case of unit magnification, no curvatures are observed in the self-imaged lines. If the magnification is bigger than unity, the light rays passing through the point in a positive or negative domain of the aperture variable contribute to the formation of the curved images. The image evaluation technique discussed here can be useful in the various applications related to the self-image formation of a parallel-line grating and it can also provide insight into the self-imaging of other periodic objects.

Frequency conversion of probe wave produced by 2D interaction with relativistic ionization front

Frequency conversion of probe electromagnetic wave induced by relativistic ionization front is theoretically analysed based on ray-tracing equations in different regimes. Downshifting as well as upshifting in frequency produced by the front is predicted. The reflected and transmitted angles can be also dramatically changed in certain cases.

Envelope equations and conservation laws describing wakefield generation and electron acceleration

Previous authors have proposed various envelope equations to describe the behavior of an electromagnetic pulse generating a wakefield. In general these retain second-order derivatives, the reason being that the eikonal contains the initial wave frequency. Here it is shown that if the evolution of the wave frequency is followed using ray-tracing equations, a first-order evolution equation is obtained. It can be shown with this formalism that wave action is conserved and the energy lost from the electromagnetic wave can be explicitly accounted for in terms of energy gained by the plasma. The energy balance equations suggest that an electron bunch which will extract energy efficiently from a wakefield can be at least as efficiently accelerated by direct interaction with the electromagnetic pulse.

WAVE PROPAGATION AND REFLECTION IN THE IONOSPHERE. AN ALTERNATIVE APPROACH FOR RAY PATH CALCULATIONS

In this work,the ray trajectory for oblique incidence is determined by solving the wave equation using the Finite Element Method (FEM). In case of ionospheric reflection the Discrete Fourier Transform (DFT),applied here in the space domain,is used to discriminate the incident and reflected waves. Ray tracing equations are combined with the Poynting vector components in order to calculate the wave trajectory taking into account the Earth's curvature. The results are illustrated for different frequencies and angles of incidence using electron density profiles obtained from the Rome digisonde station,topside profiles from the IRI-95 model and atmospheric data from the MSIS-E-90 model of NSSDC as an inputs.