Ray-theoretic Approach Research Articles

Common reflection point mapping of the mantle transition zone using recorded and 3-D synthetic ScS reverberations

SUMMARY The method of ScS reverberation migration is based on a ‘common reflection point’ analysis of multiple ScS reflections in the mantle transition zone (MTZ). We examine whether ray-theoretical traveltimes, slownesses and reflection points are sufficiently accurate for estimating the thickness H of the MTZ, defined by the distance between the 410- and 660-km phase transitions. First, we analyse ScS reverberations generated by 35 earthquakes and recorded at hundreds of seismic stations from the combined Arrays in China, Hi-NET in Japan and the Global Seismic Network. This analysis suggests that H varies by about 30 km and therefore that dynamic processes have modified the large-scale structure of the MTZ in eastern Asia and the western Pacific region. Second, we apply the same procedure to spectral-element synthetics for PREM and two 3-D models. One 3-D model incorporates degree-20 topography on the 410 and 660 discontinuities, otherwise preserving the PREM velocity model. The other model incorporates the degree-20 velocity heterogeneity of S20RTS and leaves the 410 and 660 flat. To optimize reflection point coverage, our synthetics were computed assuming a homogeneous grid of stations using 16 events, four of which are fictional. The resolved image using PREM synthetics resembles the PREM structure and indicates that the migration approach is correct. However, ScS reverberations are not as strongly sensitive to H as predicted ray-theoretically because the migration of synthetics for a model with degree-20 topography on the 410 and 660: H varies by less than 5 km in the resolved image but 10 km in the original model. In addition, the relatively strong influence of whole-mantle shear-velocity heterogeneity is evident from the migration of synthetics for the S20RTS velocity model and the broad sensitivity kernels of ScS reverberations at a period of 15 s. A ray-theoretical approach to modelling long-period ScS traveltimes appears inaccurate, at least for continental-scale regions with relatively sparse earthquake coverage. Additional modelling and comparisons with SS precursor and receiver function results should rely on 3-D waveform simulations for a variety of structures and ultimately the implementation of full wave theory.

Linearized two-hydrophone localization of a pulsed acoustic source in the presence of refraction: Theory and simulations

This paper develops an efficient three-dimensional localization method for transient acoustic sources, with uncertainty estimation, based on time differences between direct and surface-reflected arrivals at two hydrophones. The localization method accounts for refraction caused by a depth-dependent sound-speed profile using a ray-theoretic approach for calculating eigenray travel times and partial derivatives. Further, the method provides localization error estimates accounting for uncertainties of the arrival times and hydrophone locations, as well as for depth-dependent uncertainties in the sound-speed profile. In the first of two steps, source depth and range to each hydrophone are estimated using an iterative, linearized Gauss-Markov inversion scheme. In the second step, the estimated source ranges are combined with the hydrophone locations to obtain the source location in the horizontal. Localization performance is analyzed in a simulation study, and the linearized localization estimates and uncertainties are validated by comparison with a fully nonlinear (but numerically intensive) Markov-chain Monte Carlo inversion.

Bayesian linearized two-hydrophone localization of a pulsed acoustic source

A three-dimensional localization method for transient acoustic sources is developed, based on time differences between direct and surface-reflected arrivals at two hydrophones. The method accounts for refraction caused by a depth-dependent sound-speed profile using a ray-theoretic approach, and, further, it provides localization error estimates accounting for uncertainties of the arrival times and hydrophone locations, as well as for depth-dependent uncertainties in the sound-speed profile. In the first of two steps, source depth and range to each hydrophone are estimated using an iterative, linearized Gauss-Markov inversion scheme. In the second step, the estimated source ranges are combined with the hydrophone locations to obtain the source location in the horizontal. Localization performance is analyzed in a simulation study, and the linearized localization estimates and uncertainties are validated by comparison with a fully-nonlinear (but numerically intensive) Markov-chain Monte Carlo inversion. [Wor...

On the validation of seismic imaging methods: Finite frequency or ray theory?

AbstractWe investigate the merits of the more recently developed finite‐frequency approach to tomography against the more traditional and approximate ray theoretical approach for state of the art seismic models developed for western North America. To this end, we employ the spectral element method to assess the agreement between observations on real data and measurements made on synthetic seismograms predicted by the models under consideration. We check for phase delay agreement as well as waveform cross‐correlation values. Based on statistical analyses on S wave phase delay measurements, finite frequency shows an improvement over ray theory. Random sampling using cross‐correlation values identifies regions where synthetic seismograms computed with ray theory and finite‐frequency models differ the most. Our study suggests that finite‐frequency approaches to seismic imaging exhibit measurable improvement for pronounced low‐velocity anomalies such as mantle plumes.

Theory and applications of refractive index-based optical microscopy to measure protein mass transfer in spherical adsorbent particles

This work provides the theoretical foundation and a range of practical application examples of a recently developed method to measure protein mass transfer in adsorbent particles using refractive index-based optical microscopy. A ray-theoretic approach is first used to predict the behavior of light traveling through a particle during transient protein adsorption. When the protein concentration gradient in the particle is sharp, resulting in a steep refractive index gradient, the rays bend and intersect, thereby concentrating light in a sharp ring that marks the position of the adsorption front. This behavior is observed when mass transfer is dominated by pore diffusion and the adsorption isotherm is highly favorable. Applications to protein cation-exchange, hydrophobic interaction, and affinity adsorption are then considered using, as examples, the three commercial, agarose-based stationary phases SP-Sepharose-FF, Butyl Sepharose 4FF, and MabSelect. In all three cases, the method provides results that are consistent with measurements based on batch adsorption and previously published data confirming its utility for the determination of protein mass transfer kinetics under a broad range of practically relevant conditions.

Rainbows from inhomogeneous transparent spheres: a ray-theoretic approach

A ray-theoretic account of the passage of light through a radially inhomogeneous transparent sphere has been used to establish the existence of multiple primary rainbows for some refractive index profiles. The existence of such additional bows is a consequence of a sufficiently attractive potential in the interior of the drop, i.e., the refractive index gradient should be sufficiently negative there. The profiles for which this gradient is monotonically increasing do not result in this phenomenon, but nonmonotone profiles can do so, depending on the form of n. Sufficiently oscillatory profiles can lead to apparently singular behavior in the deviation angle (within the geometrical optics approximation) as well as multiple rainbows. These results also apply to systems with circular cylindrical cross sections, and may be of value in the field of rainbow refractometry.

Two-hydrophone localization of a click source in the presence of refraction

This paper introduces a ray-theoretic method for fast localization of an impulsive sound source in a general stratified ocean environment by measuring relative times of direct and surface-reflected arrivals at two hydrophones. The localization problem is solved through ray tracing and wave-front tracking in the time domain. This method is computationally efficient and allows for fast, near-real time estimation of the source location, which is a significant advantage for operational localization. The accuracy of the ray-theoretic approach is compared with that of the straight-line approximation. Further, localization errors associated with the angular and temporal discretization as well as with the variability of the hydrophone depths and the sound-speed profile are addressed.

Infrasound modeling in a spherical heterogeneous atmosphere

The deployment of a worldwide network for infrasound detection requires numerical methods for modeling these signals over long distances. A ray theoretical approach appears robust and efficient. It furthermore allows a straightforward interpretation of recorded phases. We have developed a three‐dimensional Hamiltonian ray tracing for modeling linear acoustic waves in the atmosphere. Propagating over distances superior to 500 km requires the curvature of the Earth to be considered, which is achieved by using spherical coordinates. High atmospheric winds are properly handled through a modified Hamiltonian. These winds as well as sound velocity can change significantly during long‐lasting propagations; these variations are also included in our modeling. Finally, the amplitude of infrasonic signals is computed by concomitantly solving for paraxial rays and assessing the evolution of the ray tube thus defined. We present the theory for this atmospheric infrasound modeling and some simple applications that establish its robustness and potential.

Multiparameter two‐dimensional inversion of scattered teleseismic body waves 2. Numerical examples

In this paper, we investigate the formal inversion of synthetic teleseismic P coda waves for subsurface elastic properties using a ray theoretic approach which assumes single scattering [Bostock et al., this issue]. We consider a model comprising an idealized lithospheric suture zone whose geometrical configuration is drawn from previous deep crustal seismic studies. Two‐dimensional, pseudospectral synthetic seismograms representing plane waves propagating through this model are preprocessed to extract an estimate of the scattered wave field generated by short‐wavelength structure. These data are employed in a series of numerical simulations which examine the dependence of multiparameter inversion results on a range of input parameters. In particular, we demonstrate (1) the contrasting sensitivity which forward and backscattered waves display to structural recovery, (2) the diminution of the problem null‐space accompanied by increased source coverage, (3) improvements in model reconstruction achieved through simultaneous treatment of multiple scattering modes, and (4) the robustness of the method for data sets with noise levels and receiver geometries that approach those of field experiments.

Resolving power of surface wave polarization data for higher-order heterogeneities

The resolving power of polarization data compared with that of phase data is investigated by employing both synthetic and observed data sets, using the linear relationship between the phase velocity perturbation and the phase or polarization anomaly. In order to investigate the intrinsic differences between phase and polarization data, a synthetic test is first undertaken using a white noise model with sufficiently uniform coverage of ray paths. This test shows that polarization data can retrieve higher-order heterogeneities of degrees up to 20 almost completely, despite damping and smoothing effects, whilst phase data can only retrieve those of degrees lower than 8 with reasonable damping. Next, about 4000 phase and 2500 polarization records are collected for minor- and major-arc Rayleigh waves (R1 and R2) in the frequency range 4–12 mHz. To correct the Rayleigh wave polarization data, the misorientation of each station is estimated from the polarization of long-period P waves propagating mainly in the lower mantle. The phase and polarization data are then inverted for the phase velocity distribution in spherical harmonics with degrees up to 15. The phase velocity maps derived from the phase data are quite consistent with previous studies, whilst those from the polarization data show some discrepancies. For example, the correlation between the phase and polarization models is quite good for low even degrees such as 2, 4 and 6, but not for low odd degrees or degrees higher than 8. The gradients of ampli-tude spectra from the polarization data are smaller than those from the phase data, especially at degrees higher than 6, which suggests a slightly higher sensitivity of the polarization data to higher-order heterogeneities. Nevertheless, the overall spectral characteristics of both models are similar; that is, low-order heterogeneities are dominant whilst higher orders are clearly reduced. Further investigation using a synthetic test with the same uneven paths as the observed data shows the suppression of higher-order heterogeneities. Since the synthetic test with even paths retrieves higher-order heterogeneities sufficiently, this result strongly suggests the path averaging effect of uneven ray paths that is intrinsic in the ray theoretical approach used in this study as well as almost all the global inversions. Although inversions based on geometrical ray theory have some limited resolving power with the current status of global records, polarization data are indeed helpful in resolving higher-order lateral heterogeneities with the dense and uniform path coverage that is becoming available.

Benchmarking an acoustic coupling theory for elastic shells of arbitrary shape

Coupling coefficients for thin shells of arbitrary curvature based upon a ray-theoretic approach devised by the authors are benchmarked against exact and high-frequency results for canonical geometries, i.e., the spherical and infinite circular cylindrical shells. The well-known high-frequency approximations of the coupling coefficients are found to describe their magnitude accurately but not the phase for midfrequencies (between ring and coincidence) when compared with results obtained by applying the Sommerfeld–Watson transformation to the exact modal series. The coupling coefficients characterize both the magnitude and the phase accurately and simply in terms of fundamental physical parameters.

Comparison of membrane wave coupling coefficients

Comparisons are made between various estimates of the coupling coefficients that characterize the longitudinal wave contribution to acoustic scattering from thin elastic circular cylindrical and spherical shells. A common method of obtaining the coupling coefficient is through the application of the Sommerfeld–Watson transformation to the exact modal series solution. From this result, a high-frequency approximation can be derived that accurately describes the magnitude, but it does not adequately represent the phase. These are compared to coupling coefficients determined using a ray-theoretic approach. This representation describes both the magnitude and the phase well when compared to that obtained from the exact modal series result using the Sommerfeld–Watson transformation. [Work supported by ONR.]

Scattering wave energy propagation in a random isotropic scattering medium: 1. Theory

In this paper we provide a complete formulation of scattered wave energy propagation in a random isotropic scattering medium. First, we formulate the scattered wave energy equation by extending the stationary energy transport theory studied by Wu (1985) to the time dependent case. The iterative solution of this equation gives us a general expression of temporal variation of scattered energy density at arbitrary source and receiver locations as a Neumann series expansion characterized by powers of the scattering coefficient. The first term of this series leads to the first‐order scattering formula obtained by Sato (1977). For the source and receiver coincident case, our solution gives the corrected version of high‐order formulas obtained by Gao et al. (1983b). Solving the scattered wave energy equation using a Fourier transform technique, we obtain a compact integral solution for the temporal decay of scattered wave energy which includes all multiple scattering contributions and can be easily computed numerically. Examples of this solution are presented and compared with that of the single scattering, energy flux, and diffusion models. We then discuss the energy conservation for our system by starting with our fundamental scattered wave energy equation and then demonstrating that our formulas satisfy the energy conservation when the contributions from all orders of scattering are summed up. We also generalize our scattered wave energy equations to the case of nonuniformly distributed isotropic scattering and absorption coefficients. To solve these equations, feasible numerical procedures, such as a Monte Carlo simulation scheme, are suggested. Our Monte Carlo approach to solve the wave energy equation is different from previous works (Gusev and Abubakirov, 1987; Hoshiba, 1990) based on the ray theoretical approach.

Ray-theoretical relations between intrinsic and apparent dissipation factors in the Earth

Summary. A ray-theoretical approach is successfully employed for obtaining the relations between the intrinsic dissipation factors of P- and S-waves, Qα and Qβ, and the apparent dissipation factors of toroidal and spheroidal oscillations, QTor and QSphThese are given in quite simple forms and are expressed, to zero-order approximatiom which neglects the effect of reflection of waves at an interface and/or at a free surface), as where α and β are the P- and S-wave velocities and the integrals are evaluated long the surface to surface P- and S-rays. Some numerical comparisons support the validity of these formulae for higher modes of toroidal and radial oscillations of a realistic earth model. A first-order approximation reveals that the existence of a discontinuity in an earth induces a systematic fluctuation in normal-mode Q values for any given phase velocity as a function of frequency (or radial mode number).

Prediction of ocean basin bottom reverberation using a range‐dependent reverberation model

The NRL model for underwater acoustic reverberation incorporates range dependent bathymetry and sound speed profiles. An extended ray theoretic approach for propagation estimates is combined with boundary scattering models to predict, among other measures, reverberation time histories. This paper presents model predictions of ocean basin and margin bottom reverberation for a monostatic source‐receiver geometry at a deep water site in the Northeast Pacific. The reverberation time histories are calculated for a number of radials which are selected to include prominent features such as seamounts, fracture zones, islands, and continental slopes. These time histories are azimuthally convolved with a receiver beam pattern to approximate the total basin reverberation response. The strong backscattered returns from both the continental shelf of North America and large fracture zones are of particular interest. Their strength can be 30 to 50 dB above the normal reverberation decay levels, making them a serious interference, even when located in the beam sidelobes. These predictions provide valuable information for source‐receiver design, signal selection, and pulse repetition rates to be used in future deep sea experiments. [Work supported by Naval Electronic Systems Command, Code 612.]

Ray-theoretical approach to torsional oscillations of a two-layered spherical shell and application to an inversion problem.

A ray-theoretical approach was adopted to obtain an asymptotic frequency equation for torsional oscillations of a spherically symmetrical Earth consisting of a two-layered solid shell, that is, the crust and mantle. It is shown that the equation is utilized for the inversion problem of getting the shear wave velocity distribution in the mantle from torsional eigenfrequencies. Two different methods of solving the problem are then presented. Numerical experiments for a model Earth confirm the validity of these methods and show the necessity of taking the Moho discontinuity into consideration for this kind of inversion method.

Some properties of spheroidal modes of a homogeneous elastic sphere with special reference to radial dependence of displacement

A ray-theoretical approach is attempted to interpret radial eigenfunctions (radial and tangential displacements) of spheroidal modes of a homogeneous elastic sphere. It is shown that surface displacements of the mode solutions agree well with those expected when relevant body waves are incident on the free surface. The possibility of interpreting amplitude dependence on depth in terms of interference phenomenon of up-going and down-going body waves is also demonstrated. The radial eigenfunctions are computed for a variety of spheroidal modes with relatively high radial mode numbers. An interesting feature is found in these functions: The number of node surfaces in the radial distribution of the radial displacement increases systematically for a given radial mode in conformity with a change of surface value of the tangential displacement from negative to positive with increasing colatitudinal order number.

On Trapped Modes of Rotating Fluids in Spherical Shells

The question of existence of axisymmetric, equatorially trapped modes in rotating spherical shells (Stern, 1963; Stewartson and Rickard, 1969) is approached by means of a numerical simulation. The existence of one trapped mode is confirmed, and the dependence of its frequency on the thickness of the shell is investigated. The ray theoretical approach of Bretherton (1964) is also reconsidered, and it is found that in a shell of given thickness there are only a limited number of closed ray patterns which are confined to the vicinity of the equator. A continuous band of frequencies is associated with each one of these rays. It is found that the frequencies derived by the numerical simulation for thin shells agree with the maximum frequency in the first of these bands. It is conjectured that this fact may be associated with the viscous boundary conditions driving the forced oscillations inside the shell.