Generalized Ray Method Research Articles

Generalized ray method for three-dimensional propagation in a penetrable wedge

A generalized ray solution is presented for the three-dimensional (3-D) acoustic field produced by a transient point source in a rigid-bottom, fast-speed fluid-bottom, and fast-speed elastic-bottom wedge modeling a continental shelf environment. The effect of diffraction (scattering) at the wedge apex is ignored in the solution, but nevertheless the solution is exact and complete for the image component of the field, expressed by a sum of partial waves including a pulse emitted from the source plus a finite number of pulses reflected off the wedge boundaries. Time records of the acoustic pressure illustrating 3-D acoustical propagation effects are evaluated at five receivers located at equal radial range in the horizontal from the source, but different orientation to the source, measured by the azimuthal angle assuming values: 0^{circ } (down-slope), 45^{circ } (obliquely down-slope), 90^{circ } (cross-slope), 135^{circ } (obliquely up-slope), and 180^{circ } (up-slope). The arrival times of the spherical and head waves propagating along various paths are evaluated at each receiver for each bottom. It was found that the time interval between the first arrival and the ultimate arrival diminished, and the pulses became more peaked, as the azimuthal angle of the receiver increased. For the rigid-bottom wedge, we have found that all backscattered pulses are significant at the up-slope receiver. For the fluid-bottom wedge, we have found that: at each receiver, the source-pulse is preceded by the ground wave which is much weaker than the water wave; and, at the up-slope receiver, the backscattered pulses are insignificant. For the elastic-bottom wedge, we have found that: at each receiver, the ground-wave-response begins earlier than that in the fluid-bottom wedge, and thus the record separates out into distinct ground-wave and water-wave phases; and, at the up-slope receiver, the backscattered pulses are significant.

Laser thermoelastic generation in metals above the melt threshold

An approach is presented for calculating thermoelastic generation of ultrasound in a metal plate exposed to nanosecond pulsed laser heating, sufficient to cause melting but not ablation. Detailed consideration is given to the spatial and temporal profiles of the laser pulse, penetration of the laser beam into the sample, the appearance and subsequent growth and then contraction of the melt pool, and the time dependent thermal conduction in the melt and surrounding solid throughout. The excitation of the ultrasound takes place during and shortly after the laser pulse and occurs predominantly within the thermal diffusion length of a micron or so beneath the surface. It is shown how, because of this, the output of the thermal simulations can be expressed as axially symmetric transient radial and normal surface force distributions. The epicentral displacement response to these force distributions is obtained by two methods, the one based on the elastodynamic Green's functions for plate geometry determined by the Cagniard generalized ray method and the other using a finite element numerical method. The two approaches are in very close agreement. Numerical simulations are reported on the epicentral displacement response of a 3.12 mm thick tungsten plate irradiated with a 4 ns pulsed laser beam with Gaussian spatial profile, at intensities below and above the melt threshold.

Analysis and experiment on transient dynamic response in finite mindlin plate

The transient wave propagation in the finite rectangular Mindlin plate is investigated by the analytical and experimental methods. The generalized ray method (GRM) which has been successfully applied to study the transient responses of beams, planar trusses, space frames and infinite layered media is extended to investigate the transient wave propagation and early short time transient response in finite Mindlin plate. Combining the wave solution, the shock source and the boundary conditions, the ray groups transmitted in the finite rectangular plate can be determined. Numerical simulations and experiments are performed and compared with each other. The results show that the transient wave propagation and early short time transient responses in the finite plate can be studied using the GRM. The early short time transient accelerations are very large for the finite plate subjected to the unit impulse, while the early short time transient displacements are very small. The early short time transient accelerations under the unit impulse are much larger than those under the unit step impulse. The thickness and material characteristics have remarkable effects on the early short time transient responses.

Transient wave propagation and early short time transient responses of laminated composite cylindrical shells

The generalized ray method (GRM) and the method of reverberation ray matrix (MRRM) have been successfully used to study the transient elastic wave transmission in the beams, planar trusses, space frames and infinite layered media. In this paper, the GRM and MRRM are extended to investigate the early short time transient responses of laminated composite cylindrical shells with finite size under impact load. Using the Laplace transformation, the ray groups transmitting in the laminated cylindrical shells under the shock load are yielded by means of the boundary conditions. The reverberation-ray matrix representing the multi-reflected and scattered waves in the laminated cylindrical shells with finite structural size is deduced. Using FFT algorithm, the transient response corresponding to each ray group can be derived. Through the numerical simulations, it is seen that the early short time transient accelerations of the laminated composite cylindrical shells under impact loads are very large, but the early short time transient shear strain and displacement are very small. Furthermore, the effects of the stacking sequence, thickness of the composite cylindrical shells and different shock loads on the early short time transient responses of the laminated composite cylindrical shells with finite size are also analyzed.

Early short time transient response of finite L-shaped Mindlin plate

The early short time transient response of an L-shaped plate is investigated by the method of reverberation ray matrix (MRRM) and the generalized ray method (GRM). The MRRM and GRM have been successfully used to study the transient elastic wave transmitting in the beams, planar trusses, space frames and infinite layered media. In the present paper, these two methods are extended to investigate the flexural wave propagation in a finite L-shaped Mindlin plate. The reverberation-ray matrix representing the multi-reflected and scattered waves in the L-shaped plate is presented. The rays in the GRM propagating in the L-shaped plate are determined by the shock source and the boundary conditions. From the numerical results, it indicates that the early short time transient responses of L-shaped plate calculated by the MRRM are the same as those by the GRM. The early short time transient acceleration responses are very large, while the early short time transient displacement responses are very small for the L-shaped plate subjected to the unit impulse. Furthermore, the effects of the different shock signals and the plate thickness on the early short time transient responses of the finite L-shaped Mindlin plates are also analyzed.

Transient impact responses of laminated composite cylindrical shells

The generalized ray method (GRM) has been successfully used to study the transient elastic wave transmitting in the beams, planar trusses, space frames and infinite layered media. In this letter, the GRM is extended to investigate the early short time transient responses of laminated composite cylindrical shells under impact load. By using the Laplace transformation and referring to the boundary conditions, the ray groups transmitting in the finite laminated cylindrical shells under the shock load are obtained and the transient response related to each ray group can be derived via FFT algorithm. From the numerical results, it is shown that the early short time transient accelerations of the laminated composite cylindrical shell under impact loads are very large. But the short time transient shear strain and displacement are very small.

The Earth Flattening Transformation in Body Wave Theory

Summary A simple class of power law transformations between spherical and plane models is investigated to determine the effect of the Earth flattening approximation in body wave studies. The transformations lead to the correct kinematic properties and should be chosen to reduce the inaccuracies in the dynamic properties. It is concluded that except in a fluid, the Love wave transformation of Biswas & Knopoff is the most satisfactory for SH, SV and P ray studies, Biswas's transformation for Rayleigh waves is not suitable for body wave studies, and the transformation used by Helmberger is optimum in a fluid. htrodnction The computation of theoretical seismograms in an inhomogeneous, spherically symmetric, Earth model has received considerable attention recently. Gilbert & Helmberger (1972) have applied the Cagniard-de Hoop generalized ray method to a layered sphere and the method has been used with considerable success by Helmberger & Wiggins (1971) in upper mantle studies. Gilbert & Helmberger (1972) describe their method as a direct approximation to the spherical wave functions. These approximations are invalid in layers where the ray is travelling nearly tangentially, e.g. near the turning point of the ray. If an Earth flattening approximation is applied first, the Cagniard-de Hoop method can be applied exactly to the approximate model of homogeneous plane layers. This method has been used by Miiller (1970) and is in fact the method used by Helmberger & Wiggins (1971) (Wiggins, private communication). Helmberger (1973) has used an identical transformation for velocity and density, and compared the exact solution in a homogeneous model with the Cagniardde Hoop solution. In this paper we shall discuss the nature of the approximation made in the Earth flattening approximation. In a recent paper, Hill (1972) has considered the approximations to the spherical wave functions in more detail. He obtains phase corrections to the plane wave approximations and frequency dependent corrections to the plane wave reflection coefficients. The combined effect of these approximations on the dynamic properties of the body wave and the choice of density transformation is not investigated. Chapman & Phinney (1972) have described an alternative spectral method for computing body wave seismograms. Although their method includes a solution of the fourth order spherical wave equations, it is necessary to match this to an approximate WKBJ solution. The results in this paper can also be used to indicate the approximation made here.

Generalized ray theory for an Epstein profile

The generalized ray method as extended by Heyman & Felsen is used for investigating an Epstein monotonic transition zone. Such an Epstein profile gives rise to normal turning rays, reversed turning rays and space-domain caustics. These wave species can now all be investigated using a complex integration contour in the forming of the synthetic seismogram. We present numerical results of the method and compare it to Chapman's (WKBJ seismogram) method. The waveforms generally agree well although a DC value difference is sometimes present in the seismograms. During the analysis we show that at the dark side of the caustic the arrival can be interpreted as a contribution due to tunnelling rays.

A Procedure for Modelling Shallow Dislocation Sources with Tensional Component

The source terms of a tensile dislocation are presented in terms ofthe wavenumber and circular frequency. Expressions for displacements on thesurface of a half-space due to an arbitrary oriented dislocation are alsoobtained by using the generalized ray method. The result shows that thefar-field waveforms contributed by tensile source are different from othersources. Considering that not all large earthquakes can be explainedwith the shear dislocation model, these expressions may be efficient inanalysing some earthquakes.

Transient elastic waves propagating in a multi–layered medium subjected to in–plane dynamic loadings. I. Theory

The propagation of elastic transient waves in a multi–layered medium subjected to in–plane loadings is investigated in this study. One of the objectives of this study is to develop an effective analytical method for determining transient full–field solutions in the layered medium. A matrix method is developed by expanding the matrix solution obtained directly from the boundary–value problem in the integral transform domain into a power series of the phase–related reflection and transmission matrix which characterizes the multiple reflections and transmissions of all waves in every layer. The transient response of the multi–layered medium is decomposed into infinite wave groups in which the waves are either reflected by, or transmitted through, the interfaces. The connection between the proposed matrix method and the generalized ray method is established for the layered medium in the transform domain. The matrix representation of the solution enables us to calculate the transient response of the layered medium without tracing the ray path manually. The obtained analytical solution can easily be applied to numerical calculations. The double inverse transform is performed based on Cagniard9s method and the theoretical transient solution for a layered half–space subjected to an in–plane dynamic force will be presented in part II. An experimental set–up that simulates the plane stress condition for a layered half–space was established to obtain the dynamic displacement response. The experimental result agrees well with the theoretical solution. The proposed methodology in this study can be extended to solve more complicated problems such as waves propagating in three–dimensional space.

P wave velocity of Proterozoic upper mantle beneath central and southern Asia

P wave velocity structure of Proterozoic upper mantle beneath central and southern Africa was investigated by forward modeling of Pnl waveforms from four moderate size earthquakes. The source‐receiver path of one event crosses central Africa and lies outside the African superswell while the source‐receiver paths for the other events cross Proterozoic lithosphere within southern Africa, inside the African superswell. Three observables (Pn waveshape, PL‐Pn time, and Pn/PL amplitude ratio) from the Pnl waveform were used to constrain upper mantle velocity models in a grid search procedure. For central Africa, synthetic seismograms were computed for 5880 upper mantle models using the generalized ray method and wavenumber integration; synthetic seismograms for 216 models were computed for southern Africa. Successful models were taken as those whose synthetic seismograms had similar waveshapes to the observed waveforms, as well as PL‐Pn times within 3 s of the observed times and Pn/PL amplitude ratios within 30% of the observed ratio. Successful models for central Africa yield a range of uppermost mantle velocity between 7.9 and 8.3 km s−1, velocities between 8.3 and 8.5 km s−1 at a depth of 200 km, and velocity gradients that are constant or slightly positive. For southern Africa, successful models yield uppermost mantle velocities between 8.1 and 8.3 km s−1, velocities between 7.9 and 8.4 km s−1 at a depth of 130 km, and velocity gradients between −0.001 and 0.001 s−1. Because velocity gradients are controlled strongly by structure at the bottoming depths for Pn waves, it is not easy to compare the velocity gradients obtained for central and southern Africa. For central Africa, Pn waves turn at depths of about 150–200 km, whereas for southern Africa they bottom at ∼100–150 km depth. With regard to the origin of the African superswell, our results do not have sufficient resolution to test hypotheses that invoke simple lithospheric reheating. However, our models are not consistent with explanations for the African superswell invoking extensive amounts of lithospheric thinning. If extensive lithospheric thinning had occurred beneath southern Africa, as suggested previously, then upper mantle P wave velocities beneath southern Africa would likely be lower than those in our models.

Are Chronons the Elementary Particles in space and Time?

A method for the finite element computation of the synthetic seismograms of SH wave propagation in a two-dimensional(2-D)medium is proposed. The finite element method has the advantage of specifying the arbitrary density and velocity field in the medium.To complete the waveformmodeling for an earthquake,the transformation of line source snapshotsand seismograms of the 2-D finite element modeling to the point sourceresponses is critically necessary.The accuracy of the finite element modeling is verified by comparing it with the generalized ray method and with afinite difference method which computes the 2-D wave field from a different approach.The aim of this study is to demonstrate the successfulness ofseismogram synthesis by the finite element method and to propose thepotential applications of this method in the Taiwan area.

Travel time computation in stratified anisotropic media: Generalized ray method

The generalized ray, combined with the system matrix, can be utilized to calculate the travel time of the elastic waves in stratified anisotropic media. The method proposed by Van der Hijden is modified to calculate the horizontal slowness (corresponding to travel time) of the Cagniard path. The accuracy of this method is verified by comparing the results with those of perturbation and ray tracing methods.

TUNNELLING AND THE GENERALIZED RAY METHOD IN PIECEWISE HOMOGENEOUS MEDIA1

The contributions of the tunnelled constituents to a seismic wave are analysed in two different configurations pertaining to homogeneous acoustic media: a thin high-velocity layer, present in a plane-layered configuration, and a thin layer in media separated by dipping interfaces. The generalized ray method in the far-field is used to determine them. We expand around the relevant ray parameters in order to determine the characteristics of the tunnelling ray and find that the most important feature of this type of ray is a phase (in terms of asymptotic ray theory) which has a real and an imaginary part. Numerical results illustrate this.

Acoustic waves generated by a point source in a sloping fluid layer

The propagation of acoustic waves generated by a point source in a wedge-shaped fluid layer on top of a semi-infinite space is analyzed by applying the generalized ray method. The horizontal top surface is assumed at a constant pressure and the lower surface of the wedge is in contact with another large body of fluid. Numerical results are presented for the case of a slow wave-speed bottom and a fast wave-speed bottom with three different dipping angles (0°, 5°, 10°) at various locations near a center of explosion with a step-time function. The diffraction from the apex of the wedge is neglected in this otherwise exact analysis.

Seismic Wave Propagation in Stratified Media

In order to fully appreciate this book, it is necessary to recall some of the recent history of body wave seismology. Until the late 1960s, most of our knowledge of subsurface structure came from travel time studies. Pekeris [1948] and Haskell [1953] had shown how to model seismic data, but existing computers limited the use of their methods to the computation of dispersion curves for simple earth models. Then Helmberger [1968] used the Cagniard‐de Hoop method [de Hoop, 1960] to model refraction arrivals and thereby demonstrated the practicality of seismic modeling in the time domain. The Cagniard‐de Hoop method is a generalized ray method, which means (in practical terms) that it is good for synthesizing first motions but not so good for the later parts of the seismogram. Accordingly, Fuchs and Muller [1971] returned to the methods of Pekeris and Haskell and showed that with large modern computers, the whole seismogram could be synthesized. However, problems remained, because Haskell matrices are numerically unstable when used to synthesize SV body waves. Methods of overcoming this instability were found (and are still being found), but they are all, with the possible exception of the methods of Schmidt and Tango [1986] and of Chin et al. [1984], complicated, difficult to program, and lacking in physical insight.

Finite-difference seismograms for SH waves

The accuracy of the finite-difference method for generating synthetic seismograms of SH wave propagation in cylindrically symmetric media is discussed. The finite-difference method has the advantage that arbitrary density and velocity fields in the medium may be specified. A point source is generated by a simple transformation of a line source. The accuracy of the finite-difference seismograms in flat- and dipping-layered media is confirmed by comparison with the generalized ray method. A source radiation pattern is inserted by introducing a “near-field” which has permanent displacement near the source. Strong motion synthetics are constructed with this new method for the 1968 Borrego Mountain earthquake as recorded at El Centro. Good fits to the data are achieved using the laterally varying model determined by a detailed refraction survey and the source parameters determined by teleseismic waveform modeling. Shallow faulting is no longer necessary to explain the long-period surface-wave development.

Synthetic seismograms

The interpretation of seismic data, in particular oceanic refraction data, has improved considerably, in recent years using synthetic seismograms. Several different methods have been developed for computing synthetic seismograms and within the limitations of the various approximations, shown to give equivalent results. The computation of theoretical seismograms can be divided into two parts: the evaluation of the response in a transform domain (normally frequency and horizontal slowness), and the evaluation of the inverse transforms (Fourier, Fourier-Bessel, or Legendre transforms). Developments of the Haskell matrix method are widely used to evaluate the transformed response. Problems with numerical stability have been solved, and the method has been extended to use asymptotic solutions in inhomogeneous layers. Often the complete response is expanded, at least at some interfaces, into “rays” and only a limited set of rays investigated. The ray expansion often simplifies the evaluation of the inverse transforms and takes advantage of separation in the time domain. The inverse transform methods fall into two catagories: spectral and slowness methods. In spectral methods, the inverse wavenumber transform is evaluated first and the intermediate result is the spectral response for a point source. In slowness methods, the inverse frequency transform is evaluated first and the intermediate result is the impulse “plane wave” response. For a general response function, neither method is numerically preferable but the intermediate result in the slowness method is convenient as it corresponds to slant stacking the data. With the ray expansion, the slowness method is often preferable as the inverse transforms can be evaluated exactly—the generalized ray method and the WKBJ seismogram. The last method has recently been extended to laterally inhomogeneous media.

On impulsive wave propagation in a spherically symmetric model

Slowness methods have been widely used to study impulsive wave propagation and to compute synthetic seismograms. The generalized ray method and WKBJ seismograms are examples. When applied to a spherically symmetric model, asymptotic or numerical approximations have been used for the Legendre function. In this note it is pointed out that an analytic inverse transform exists for the Legendre function. This can be used when the waveslowness integral is real but cannot be used in the generalized ray method.

Exact and Approximate Generalized Ray Theory in Vertically Inhomogeneous Media

Summary The generalized ray method in a vertically inhomogeneous model is formulated without any approximation by homogeneous layers. The solution is obtained as an infinite series in multiply ‘ reflected ’ waves. Each term can be solved using the exact method or the plane-wave, firstmotion or geometrical approximations. It is shown that the firstmotion approximation of the series converges rapidly, the ratio of successive terms in the infinite series being - (21+ 1)(21)(6/n)’. In addition it is shown that the first-motion approximation, which reduces to the geometrical approximation when the latter is valid, is a useful alternative to geometrical ray theory, being more generally valid and being almost as simple to compute.