Geometrical Ray Theory Research Articles

Shadows of structured beams in lenslike media.

The self-healing phenomenon of structured light beams has been comprehensively investigated for its important role in various applications including optical tweezing, superresolution imaging, and optical communication. However, for different structured beams, there are different explanations for the self-healing effect, and a unified theory has not yet been formed. Here we report both theoretically and experimentally a study of the self-healing effect of structured beams in lenslike media, this is, inhomogeneous lenslike media with a quadratic gradient index. By observing the appearance of a number of shadows of obstructed structured wave fields it has been demonstrated that their self-healing in inhomogeneous media are the result of superposition of fundamental traveling waves. We have found that self-healing of structured beams occurs in this medium and, interestingly enough, that the shadows created in the process present sinusoidal propagating characteristics as determined by the geometrical ray theory in lenslike media. This work provides what we believe to be a new inhomogenous environment to explain the self-healing effect and is expected to deepen understanding of the physical mechanism.

Near-infrared star map simulation for starlight refraction sensor based on ray tracing

The application of starlight refraction navigation to spacecraft and space weapons is a significant development direction. Due to the huge cost of spacecraft experiments, the functional validation of this technology relies mainly on ground simulations. The existing refraction star-map simulation ignores the starlight refraction angle’s change with time and position, which cannot meet the engineering requirements. Based on the timed/saber dataset and geometric ray theory, this paper presents a ray tracing algorithm to calculate the starlight refraction angle in different propagation paths. Combined with the coordinate conversion and energy transfer, a near-infrared refraction star-map simulation method is proposed. An all-time star sensor is used for simulation, which works in the 900–1700 nm band and can suppress strong limb background. The experimental data verify the refraction angle and energy model of the proposed method. Through the comparison between the measured data and simulated star maps, the starlight refraction angle’s bias error of the proposed method is −0.05″, which is typically smaller than −3.30″ of the traditional method, and the average flux error is 11%. The evaluation and verification outcome support the proposed star-map simulation method and can provide technical support for the optimization design, capability estimation, and algorithm validation of the starlight refraction navigation.

Born modeling for heterogeneous media using the Gaussian beam summation based Green's function

Born approximation is a commonly used approximation in the simulation of seismic wave propagation. Calculation of the Green's function in Born approximation integral is essential for Born modeling. We derive a new Born formula based on the Gaussian beam representations of Green's functions. This procedure can be used to mitigate the problems like the caustic, shadow region, and multivalued traveltime caused by multipathing that traditional geometric ray theory cannot deal with. However, due to the characteristic of complex traveltime in the Gaussian beam, we present a new isochronous stack method for Gaussian beam based Born modeling. Additionally, two basic issues, background velocity and integral region selection, are discussed. Numerical results demonstrate the accuracy and efficiency of the Gaussian beam based Born theory and implementation.

Floquet mode phenomena over 1-D infinite array illuminated by a CSB for reflectarray applications

For reflectarray antenna applications, radiation field of the one-dimensional infinite array which is illuminated by a complex source beam (CSB) is analyzed in this paper. The CSB is an excellent basis function to expand any slowly varying radiation field of reflectarray’s feed. By analytically continuing the source which is used to illuminate reflectarray from real space to complex space, CSB is generated simply. By adopting Poisson transformation, the slowly convergent superposition of individual radiation field from reflectarray is transformed to terms of Floquet modes. Asymptotic evaluation technique is utilized to calculate the Floquet mode integral in closed form. The propagation characteristic of Floquet mode follows the geometric optic ray theory in complex space. Numerical results are presented to demonstrate this approach.

DISRAY: A distributed ray tracing by map-reduce

Seismic ray tracing is one of the most important tools in seismology which is required in many applications such as body wave tomography and exploration seismology. Due to the large amount of data introduced in global tomography, most studies are still using one-dimensional (1D) ray tracing methods. Furthermore, since three-dimensional (3D) global studies generally use initial models obtained by 1D ray tracing algorithms, such algorithms are essentially desired. Regarding new advances in High Performance Computing (HPC) as well as the importance of 1D models in global tomography, this study tries to improve the 1D ray tracing algorithm through an innovative programming paradigm, Map-Reduce. Based on this, we present a novel algorithm, DISRAY, for the distributed implementation of 1D ray tracing shooting methods. It effectively decreases computational effort, and therefore accuracy could be improved. The algorithm implements the shooting method based on the geometrical ray theory inside a radially homogeneous earth model. Travel times, epicentral distances, and ray paths of the first arrival phases and other later phases from take-off angles 0° to 180° are calculated. This algorithm can efficiently be extended to more complex models. The proposed algorithm is well remarked for its efficiency and simplicity.

Probing local wind and temperature structure using infrasound from Volcan Villarrica (Chile)

We use the continuous and intense (∼107W) infrasound produced by Volcan Villarrica (Chile) to invert for the local dynamic wind and temperature structure of the atmosphere. Infrasound arrays deployed in March 2011 at the summit (2826 m) and on the NNW flank (∼8 km distant at 825 m) were used to track infrasound propagation times and signal power. We model an atmosphere with vertically varying temperature and horizontal winds and use propagation times (ranging from 23 to 24 s) to invert for horizontal slowness (2.75–2.94 s/km) and average effective sound speeds (328–346 m/s) for NNW propagating infrasound. The corresponding ratio of recorded acoustic power at proximal versus distal arrays was also variable (ranging between 0.15 to 1.5 for the peak 0.33–1 Hz infrasound band). Through application of geometrical ray theory in a uniform gradient atmosphere, these ‘amplification factors’ are modeled by effective sound speed lapse rates ranging from −15 to +4 m/s per km. NNW‐projected wind speeds ranging from −20 m/s to +20 m/s at 2826 m and wind gradients ranging from −11 to +10 m/s per km are inferred from the difference between effective sound speed profiles and adiabatic sound speeds derived from local temperature observations. The sense of these winds is in general agreement with regional meteorological observations recorded with radiosondes. We suggest that infrasound probing can provide useful spatially averaged estimates of atmospheric wind structure that has application for both meteorological observation and volcanological plume dispersal modeling.

Nonlinear and diffraction effects in propagation of N-waves in randomly inhomogeneous moving media

Finite amplitude acoustic wave propagation through atmospheric turbulence is modeled using a Khokhlov-Zabolotskaya-Kuznetsov (KZK)-type equation. The equation accounts for the combined effects of nonlinearity, diffraction, absorption, and vectorial inhomogeneities of the medium. A numerical algorithm is developed which uses a shock capturing scheme to reduce the number of temporal grid points. The inhomogeneous medium is modeled using random Fourier modes technique. Propagation of N-waves through the medium produces regions of focusing and defocusing that is consistent with geometrical ray theory. However, differences up to ten wavelengths are observed in the locations of fist foci. Nonlinear effects are shown to enhance local focusing, increase the maximum peak pressure (up to 60%), and decrease the shock rise time (about 30 times). Although the peak pressure increases and the rise time decreases in focal regions, statistical analysis across the entire wavefront at a distance 120 wavelengths from the source indicates that turbulence: decreases the mean time-of-flight by 15% of a pulse duration, decreases the mean peak pressure by 6%, and increases the mean rise time by almost 100%. The peak pressure and the arrival time are primarily governed by large scale inhomogeneities, while the rise time is also sensitive to small scales.

Wind turbine noise propagation: Comparison of measurements and predictions by a method based on geometrical ray theory

The objective of the present work has been to investigate the accuracy of the Nord2000 propagation model when used to predict noise from high sources like wind turbines. The model is representative to a new generation of �engineering� models developed on basis of analytical and heuristic solutions. The accuracy is investigated by comparison with measurements. Three experiments have been carried out for this purpose: 1) downwind and upwind propagation over flat grass-covered ground from an elevated loudspeaker, 2) downwind and upwind propagation over non-flat ground from a loudspeaker placed on top of a wind turbine nacelle, and 3) downwind propagation over non-flat ground from a single wind turbine. Experiment 1) showed for downwind propagation a very fine agreement between measured and predicted A-weighted noise levels in the considered range of propagation distances (1500 m). The bias error was 0.1 dB with a standard uncertainty of 0.7 dB. The agreement for upwind propagation was not as good. In experiment 2) the accuracy was slightly reduced compared to the downwind results of experiment 1) with a bias error of 0.5 dB and a standard uncertainty of 1.9 dB. In experiment 3) where a wind turbine was used as source the agreement was less than seen in the loudspeaker experiments.

Estimates of source parameters of M4.9 Kharsali earthquake using waveform modelling

This paper presents the computation of time series of the 22 July 2007 M 4.9 Kharsali earthquake. It occurred close to the Main Central Thrust (MCT) where seismic gap exists. The main shock and 17 aftershocks were located by closely spaced eleven seismograph stations in a network that involved VSAT based real-time seismic monitoring. The largest aftershock of M 3.5 and other aftershocks occurred within a small volume of 4 × 4 km horizontal extent and between depths of 10 and 14 km. The values of seismic moment (M ∘) determined using P-wave spectra and Brune’s model based on f 2 spectral shape ranges from 1018 to 1023 dyne-cm. The initial aftershocks occurred at greater depth compared to the later aftershocks. The time series of ground motion have been computed for recording sites using geometric ray theory and Green’s function approach. The method for computing time series consists in integrating the far-field contributions of Green’s function for a number of distributed point source. The generated waveforms have been compared with the observed ones. It has been inferred that the Kharsali earthquake occurred due to a northerly dipping low angle thrust fault at a depth of 14 km taking strike N279°E, dip 14° and rake 117°. There are two regions on the fault surface which have larger slip amplitudes (asperities) and the rupture which has been considered as circular in nature initiated from the asperity at a greater depth shifting gradually upwards. The two asperities cover only 10% of the total area of the causative fault plane. However, detailed seismic imaging of these two asperities can be corroborated with structural heterogeneities associated with causative fault to understand how seismogenesis is influenced by strong or weak structural barriers in the region.

Depth-dependent attenuation structure of the inner core inferred from short-period Hi-net data

Depth-dependent attenuation structure in the inner core below the northeast Pacific is studied utilizing the short-period Hi-net data. The commonly used method of estimating inner core attenuation (differential spectral slope analysis) is not suitable because it is too sensitive to the presence of small amplitude phases. Instead, we found that the method of relative and differential amplitudes is more robust estimator of inner core attenuation. We use the relative amplitude of PKIKP (DF) to PKPs (BC or AB) around 1.0 Hz to estimate t ∗ as a function of the epicentral distance. Geometrical ray theory was employed to correct the source and propagation effects except that of inner core attenuation. The resulting t ∗ for South American events shows a clear peak around an epicentral distance of 152 ° , whose corresponding bottoming depth is about 300 km below the inner core boundary (ICB). This pattern can be explained by a low Q P (<200), high attenuation region in a depth range of 200–300 km. We also found that, in this depth range, there exists positive correlation between attenuation and velocity, indicating seismic scattering due to anisotropic structure of iron crystals at the intermediate depth of the inner core.

Difficulties with computing anelastic plane-wave reflection and transmission coefficients

SUMMARY Consider a plane homogeneous wave incident upon an interface between two anelastic halfspaces. Computing the plane-wave displacement reflection and transmission coefficients requires determining the proper signs of the vertical slownesses of all the reflected and transmitted waves. In certain cases, this is not straightforward. Previous work has shown that choosing the signs by applying the elastic radiation condition results in certain vertical slownesses, and hence coefficients, varying discontinuously and unphysically with the angle of incidence, and that choosing the signs so that the vertical slownesses vary continuously can also produce errors. We suggest three approaches for treating these cases. In the first approach, the signs are chosen so that the vertical slownesses vary continuously up to a certain angle of incidence (close to the elastic critical angle), with the elastic radiation condition being applied beyond that. We show that this is actually just an extension of the elastic radiation condition to complex-valued squared slownesses. In the spherical wave case in which a point source lies in the upper half-space, the approach also agrees with the results obtained by applying the saddle point method to approximate the integral for the reflected wavefield. This approximation is just geometrical ray theory. The first approach also produces coefficients with unphysical aspects, but these are confined to the critical zone, where geometrical ray theory is not valid anyway. Outside of this zone, the coefficients compare quite well with the elastic ones. In the second approach, real values of the horizontal slowness are used to compute the coefficients. This results in coefficients which are continuous and have no unphysical aspects, but do not compare as well with the elastic ones. The third approach involves using other paths in the complex plane (of horizontal slowness) to evaluate the coefficients. We show two examples. The first example is an approximation which involves replacing certain vertical slownesses with their complex conjugates before the coefficients are computed. This is equivalent to evaluating the horizontal slowness along a curve just above the real axis. The second example involves using a path which yields coefficients that are almost identical to the elastic ones. The third approach also results in continuous coefficients with no unphysical aspects, which compare well with the elastic ones. The errors mentioned above suggest to us that the problem of how to correctly compute plane-wave anelastic reflection and transmission coefficients, even in the apparently simple case of a homogeneous incident plane wave, still requires further study.

Generalized Born scattering of elastic waves in 3-D media

It is well known that when a seismic wave propagates through an elastic medium with gradients in the parameters which describe it (e.g. slowness and density), energy is scattered from the incident wave generating low-frequency partial reflections. Many approximate solutions to the wave equation, e.g. geometrical ray theory (GRT), Maslov theory and Gaussian beams, do not model these signals. The problem of describing partial reflections in 1-D media has been extensively studied in the seismic literature and considerable progress has been made using iterative techniques based on WKBJ, Airy or Langer type ansatze. In this paper we derive a first-order scattering formalism to describe partial reflections in 3-D media. The correction term describing the scattered energy is developed as a volume integral over terms dependent upon the first spatial derivatives (gradients) of the parameters describing the medium and the solution. The relationship we derive could, in principle, be used as the basis for an iterative scheme but the computational expense, particularly for elastic media, will usually prohibit this approach. The result we obtain is closely related to the usual Born approximation, but differs in that the scattering term is not derived from a perturbation to a background model, but rather from the error in an approximate Green's function. We examine analytically the relationship between the results produced by the new formalism and the usual Born approximation for a medium which has no long-wavelength heterogeneities. We show that in such a case the two methods agree approximately as expected, but that in a media with heterogeneities of all wavelengths the new gradient scattering formalism is superior. We establish analytically the connection between the formalism developed here and the iterative approach based on the WKBJ solution which has been used previously in 1-D media. Numerical examples are shown to illustrate the examples discussed.

表面波インバージョンによる上部マントル内部の推定

A variety of methods of surface wave inversion, which enables us to investigate detailed images of the upper mantle on a regional scale, are reviewed. The study of surface wave tomography beginning in the 1980's has brought us with a significant jump in our understanding of the Earth's interior, particularly the upper mantle. Most of the studies of surface wave tomography have been based upon a geometrical ray theory, working with either dispersion curves or waveforms of surface waves. Such a simple representation of surface wave propagation has allowed us to treat a greater number of data sets, which are indispensable for obtaining high resolution tomography models. However, the ray theory, which is relying upon the high-frequency approximation, is no longer valid when the scale-length of heterogeneity is comparable to the wavelength of waves to be considered. The effects of finite frequency are particularly important for the higher-frequency surface waves, which mainly sample the crust and uppermost mantle where very strong lateral heterogeneity is likely to exist. Recent development of the 3-D sensitivity kernels allows us to treat the effects of finite frequency in the tomographic inversions. The use of such finite frequency theory will further advance the methods of surface wave tomography.

On sensitivity kernels for ‘wave-equation’ transmission tomography

SUMMARY We combine seismological scattering theory with the theory of distributions to study some properties of sensitivity kernels for finite frequency seismic delay times. The theory to be used for calculating the kernels depends on the way the measurements are made. For example, the sensitivity to the traveltime proper, that is the time to instantaneous onset of the phase arrival as defined in geometrical ray theory, has a non-zero value on the ray and zero elsewhere, whereas measurements of the smooth later part of the wave excitation can have more complicated kernels. Analysis based upon the Born approximation reveals that the behaviour of such kernels is determined by the regularization of the nth derivative of the Dirac delta supported on the unperturbed source-receiver ray, where n is the spatial dimension. Such regularization induces approximately the desired band-limitation and the associated Fresnel zone averaging. If the regularization is symmetric about the singular support of the delta its odd order derivatives vanish on this support, but the even orders render non-zero values. This explains why 3-D finite frequency kernels for body wave delay times seem—in specific, rather restrictive circumstances—to have zero sensitivity on the ray (as in the so called ‘banana-doughnut kernels’). Indeed, a regularization of the kernel can have a zero on the unperturbed ray, in the absence of caustics (that is, the medium must be simple or quasi-homogeneous) and only if the exact source signature (e.g., source time function) is known and used; both conditions can be met in synthetic cases but not—in general—in applications to real data and heterogeneous media. In general, one will not know where the oscillatory kernels have zero values, which has implications for the use of such kernels for the linearization of transmission tomography. We briefly clarify these observations by invoking a multiresolution analysis of sensitivity kernels, connected with a time-frequency analysis of phase arrival times.

Sneak Previews

In the Tutorial section of the May-June issue of Geophysics, Spetzler and Snieder provide a detailed discussion about the limitations of geometric ray theory in imaging methods applied to finite-frequency wavefield recordings. The authors show that the resolution in tomographic images is not limited to the Fresnel volume width. This conclusion might have important implications for estimating resolution limits of tomographic images.

The Fresnel volume and transmitted waves

In seismic imaging experiments, it is common to use a geometric ray theory that is an asymptotic solution of the wave equation in the high‐frequency limit. Consequently, it is assumed that waves propagate along infinitely narrow lines through space, called rays, that join the source and receiver. In reality, recorded waves have a finite‐frequency content. The band limitation of waves implies that the propagation of waves is extended to a finite volume of space around the geometrical ray path. This volume is called the Fresnel volume. In this tutorial, we introduce the physics of the Fresnel volume and we present a solution of the wave equation that accounts for the band limitation of waves. The finite‐frequency wave theory specifies sensitivity kernels that linearly relate the traveltime and amplitude of band‐limited transmitted and reflected waves to slowness variations in the earth. The Fresnel zone and the finite‐frequency sensitivity kernels are closely connected through the concept of constructive interference of waves. The finite‐frequency wave theory leads to the counterintuitive result that a pointlike velocity perturbation placed on the geometric ray in three dimensions does not cause a perturbation of the phase of the wavefield. Also, it turns out that Fermat's theorem in the context of geometric ray theory is a special case of the finite‐frequency wave theory in the limit of infinite frequency. Last, we address the misconception that the width of the Fresnel volume limits the resolution in imaging experiments.

A simple formula for evaluating the acoustic effect of balconies in protecting dwellings against road traffic noise

This study explores the noise reducing effect of a balcony and describes the development of a simple theory pertaining to the propagation of traffic noise from a road into a balcony. A new methodology is proposed that is based on the well-known prediction scheme—“Calculation of Road Traffic Noise” (CRTN)—developed in the UK. A geometrical ray theory is developed for the prediction of noise levels inside a balcony due to road traffic. The source level of road traffic noise is obtained as per the standard CRTN methodology. However, road sub-segmentations and new approaches for the prediction of noise levels at illuminated and shadow zones inside a balcony are proposed. Field measurements have been conducted on four different types of balcony to validate the proposed methodology. The insertion loss, defined as the difference in the noise levels with and without the presence of a balcony, has been used to assess the shielding effectiveness of a balcony against road traffic noise. The simple theory is validated by outdoor field measurements. It is also found that a properly designed balcony can provide considerable screening effects in protecting dwellings against road traffic noise.

Sensitivity kernels for shear wave splitting in transverse isotropic media

SUMMARY Shear wave splitting observations are often used to constrain upper-mantle anisotropy. However, their interpretation, which is still controversial, always implicitly relies on geometric ray theory. This high-frequency approximation does not take the finite frequency of the shear wavefield into account. In this paper, the effects of transverse isotropic perturbations of the elastic parameters on shear wave splitting are investigated. The theory, based on the first-order Born approximation, allows the derivation of 2-D and 3-D Frechet or sensitivity kernels for a new seismic observable called the splitting intensity. It is shown that the splitting intensity measured at the surface can be written to first order as a weighted average of two anisotropic perturbation parameters γc and γs related to the anisotropic parameter γ introduced by Mensch & Rasolofosaon. We also demonstrate, using a stationary-phase approximation, that the volume integral of the sensitivity kernels for a homogeneous transverse isotropic medium leads to the ray theoretical prediction for the splitting intensity. A numerical experiment on a two-block model demonstrates that lateral variations of anisotropy in the upper mantle introduce complicated and counterintuitive effects on shear wave splitting observed at the surface.

Global surface wave diffraction tomography

We determine the effect of replacing geometrical ray theory in surface wave tomography with scattering theory. We describe a tomographic method based on a simplified version of the scattering sensitivity kernels that emerge from the Born or Rytov approximations in which surface wave travel times are a weighted average of phase or group slowness over the first Fresnel zone of the wave. We apply this “diffraction tomography” to Rayleigh and Love wave group velocity measurements to produce group velocity maps from 20 to 150 s period on a 2° × 2° grid globally. Using identical data and damping parameters, we also produce maps using “Gaussian tomography” which is based on ray theory with intuitive Gaussian smoothing constraints. Significant differences in the amplitude and geometry of the imaged features appear primarily at long periods but exist even in the short‐period maps in regions where average path lengths are large. Diffraction tomography, therefore, is significant in most oceanic regions at all periods, but it is also important on continents at long periods at least. On average, diffraction tomography produces larger velocity anomalies in a period‐dependent band of spherical harmonic degrees, and diffraction and Gaussian tomography maps decorrelate past a critical spherical harmonic degree that also depends on period. The widths of resolving kernels that emerge from diffraction tomography are systematically larger than those from Gaussian tomography. Finally, mantle features inferred from diffraction tomography tend to have larger amplitudes and extend deeper than those from Gaussian tomography.

Determination of the influence zone for surface wave paths

SUMMARY An approximate description of the zone of influence around the propagation path for a surface wave is provided by investigating the Fresnel zones for the frequency range of interest. The influence zone about surface wave paths, over which surface waves are coherent in phase, is identified as approximately one-third of the width of the first Fresnel zone. A technique called Fresnel-area ray tracing (FRT) for surface waves has been used to estimate this region around the ray path for each frequency. The FRT technique is developed by combining two standard ray tracing methods, i.e. kinematic ray tracing (KRT) and dynamic ray tracing (DRT). To obtain the exact Fresnel area in a laterally heterogeneous structure would require the solution of a large number of KRT equations. In contrast, the FRT approach requires just a few ray tracing calculations. In the first step, the trajectory of the surface wave is computed by solving the KRT system for the phase-velocity distribution at the required frequency. In the next step, the behaviour of rays in the zone surrounding the KRT path is calculated by solving the DRT system twice; once from the source to the receiver and once more from the receiver to the source along the same trajectory. Finally, combining the solutions of these ray tracing systems using paraxial ray theory, the Fresnel area around a central ray can be estimated. Using FRT, stationary-phase fields can be constructed around a central ray path in a laterally heterogeneous structure. The influence zone around the ray path is then estimated from the stationary-phase function with simple assumptions concerning the perturbed wavefield. The estimate of the influence zone can be efficiently calculated in laterally heterogeneous structure by using the FRT technique, and allows an extension of current methods of surface wave analysis, which have commonly been based on geometrical ray theory and on the approximation of great-circle propagation. This approach allows the treatment of finite-width rays as well as deviations in propagation from the great circle induced by moderate lateral heterogeneity as revealed by recent tomography models. Such finite-width rays should be of major benefit in enhancing ray-based surface wave tomography.