Modeling Of Rayleigh Waves Research Articles

Quantitative Evaluation of Human Lens and Lens Capsule Elasticity by Optical Coherence Elastography Based on a Rayleigh Wave Model.

Evaluating the biomechanical properties of the lens and lens capsule is important for the clinical diagnosis and treatment of age-related cataracts and presbyopia. In this study, we developed an optical coherent elastography technique to assess the elasticity of the lens and lens capsule in the human eye. With age, the mean Young's modulus of the lens increased from 12.28 ± 0.87 kPa to 18.59 ± 1.45 kPa, and the lens capsule increased from 6.33 ± 0.36 kPa to 13.33 ± 0.74 kPa. The results showed that the Young's modulus of the lens capsule and lens increased with age, with the Young's modulus of the lens significantly higher than that of the lens capsule. This study reports the assessment of the elasticity of the human lens and lens capsule by the OCE technique, indicating that it may provide a potential clinical tool for advancing research on diseases affecting the lens.

An improved acoustic/elastic interface approach for 2D staggered grid finite-difference modeling of Rayleigh waves in the presence of surface topography

Rayleigh waves are generated along the free-surface and their propagation can be strongly influenced by surface topography. Therefore, a critical aspect of Rayleigh-wave study is to understand the wave-propagation behavior in the presence of surface topography. The acoustic/elastic boundary approach was incorporated into a ‘stair-case’ mesh for Rayleigh-wave modeling in the presence of surface topography. The conventional acoustic/elastic boundary approach, however, fails to completely fulfill the free-surface boundary condition at some grid nodes, leading to unphysical simulation results. To solve this problem, we develop a stable acoustic/elastic boundary approach that fully satisfies the free-surface boundary condition by adjusting the free boundary conditions at problematic nodes in the mesh. The accuracy is demonstrated by modeling tests in two-dimensional isotropic models in the presence of surface topography. And compared to the improved vacuum formulation, the proposed method takes less time. Our approach optimizes the free-surface boundary condition and enables the high-precision numerical simulation of Rayleigh waves.

An analytic model of Rayleigh waves transference in a fluid containing inhomogeneity as a circular function of depth

An analytic model of Rayleigh waves transference in a fluid containing inhomogeneity as a circular function of depth

The axisymmetric Rayleigh waves in a semi-infinite elastic solid

ABSTRACT It is well-known that Rayleigh wave, also known as surface acoustic wave (SAW), solutions in semi-infinite solids are plane waves with signatory properties like the distinct velocity and exponentially decaying deformation in the depth. Applications of Rayleigh waves are focused on the deformation and energy in the vicinity of surface of solids and less loss in the propagation. A generalized model of Rayleigh waves in axisymmetric mode is established and solutions are obtained with cylindrical coordinates. It is found that the Rayleigh waves also propagate in the axisymmetric mode with slow decay in radius, confirming the existence of surface acoustic waves is irrelevant to coordinate system. On the other hand, the solutions can be treated as plane waves in regions far away from the source. Furthermore, the particle trajectory of axisymmetric SAW is a line with constant slope rather than the signatory ellipse in Cartesian coordinate case.

Non-paraxial multi-Gaussian beam model of Leaky Rayleigh waves generated by a focused immersion transducer

A non-paraxial multi-Gaussian beam (NMGB) model is proposed for Leaky Rayleigh Waves (LRWs) generated by a focused immersion transducer at oblique incidence. Using the NMGB model, the velocity fields are calculated and compared with the corresponding results obtained by the paraxial multi-Gaussian beam (MGB) model and the more exact Rayleigh–Sommerfeld integral (RSI) model. Numerical results show that the LRW beam behavior obtained using the NMGB model agrees well with that using the RSI model, but the NMGB model is much more efficient. Moreover the NMGB model overcomes the accuracy limitation of the MGB model. Good agreement between the NMGB model and experimental measurements for both on-axis and off-axis fields is obtained when an attenuation coefficient is introduced. In addition, this model can be used to measure the attenuation coefficient with consideration of the diffraction attenuation. It is observed that the attenuation coefficient of the LRW will increase when the acoustic impendence differences between the solid and fluid decrease. The NMGB model described in this article provides an efficient tool for calculating the velocity fields of the LRW and is therefore significant for practical applications of ultrasonic measurements.

Inversion of Meteor Rayleigh Waves on Earth and Modeling of Air Coupled Rayleigh Waves on Mars

Meteor impacts and/or meteor events generate body and surface seismic waves on the surface of a planet. When meteoroids burst in the atmosphere, they generate shock waves that subsequently convert into acoustic waves in the atmosphere and seismic waves in the ground. This effect can be modeled as the amplitude of Rayleigh and other Spheroidal modes excitation, due to atmospheric/ground coupling effects. First, an inversion of the seismic source of Chelyabinsk superbolide is performed. We develop an approach in order to model a line source in the atmosphere, corresponding to the consecutive generation of shock waves by the interaction with the atmosphere. The model is based on the known trajectory. We calculate the synthetic seismograms of Rayleigh waves associated with the event by the summation of normal modes of a model of the solid part and the atmosphere of the planet. Through an inversion technique based on singular value decomposition, we perform a full Rayleigh wave inversion and we provide solutions for the moment magnitude. SEIS will likely detect seismic waves generated by impacts and the later might be further located by remote sensing differential processing. In the case of Mars, we use the same method to obtain waveforms associated with impacts on the planetary surface or in low altitudes in the Martian atmosphere. We show that the contribution of the fundamental spheroidal solid mode is dominating the waveforms, compared to that of the first two overtones. We perform an amplitude comparison and we show that small impactors (diameter of 0.5 to 2 m), can be detected by the SEIS VBB seismometer of InSight mission, even in short epicentral distances, in the higher frequencies of the Rayleigh waves. We perform an analysis based on impact rate estimations and we calculate the number of detectable events of 1 meter diameter meteor impacts to be 6.7 to 13.4 per 1 Martian year for a $Q=500$ .

Development and numerical tests of a Bayesian approach to inferring shallow velocity structures using microtremor arrays

We propose an empirical Bayesian approach to inferring shallow (depth ranges from a few to several tens of metres) S-wave velocity structures using microtremor arrays and execute numerical tests to assess the feasibility of this approach. In our approach, the estimate of the S-wave structure (posterior) is derived from an empirical S-wave structure model (prior) and phase velocities of Rayleigh waves obtained with microtremor arrays. In other words, we aim to find a model that is close to the empirical model and is able to explain phase velocities with a 1D surface-wave theory. The inversion is stabilised by the constraints from the prior model so that model parameterisation with many thin layers can be adopted. The velocity structure is individually estimated for each of two cases (assumptions): the case where we assume fundamental-mode dominance and the case where we take into account the higher modes. Optimal values of the model parameters (e.g. a thickness parameter) are found, based on Akaike’s Bayesian Information Criterion (ABIC), and the choice of the better assumption of the surface-wave theory is also based on ABIC. Numerical tests, where synthetic data is generated from a horizontally stratified two-layer model, indicate that the relative weight between a prior model and the observed data is appropriately adjusted by ABIC. It is revealed that a value of the thickness parameter required to reproduce the given two-layer model is successfully found by ABIC. We also suggest that we can make a plausible choice of the assumption of the surface-wave theory with ABIC, unless observation error is extremely large.

Nonlocal scale effect on Rayleigh wave propagation in porous fluid-saturated materials

In this paper, a Rayleigh wave model in fluid saturated porous materials based on a nonlocal Biot theory is proposed. The general characteristic equations expressed in terms of the Rayleigh wavenumbers are obtained. A specific fluid saturated porous material is used for numerical analysis. The present study shows that the nonlocal parameter does not have significant influence on the characteristics of Rayleigh waves within a low frequency range when comparing with the prediction of using the classical Biot theory. However, the influence of the nonlocal scale effect on the Rayleigh wave velocity and the displacement field becomes stronger as the response frequency increases. When the frequency exceeds a critical value, the Rayleigh wave velocity exhibits a negative dispersion. The displacement fields induced by the Rayleigh wave propagating in porous materials are also presented. An interesting phenomenon of the displacement fields is observed that the major axis of the displacement field ellipse will have a contra-rotation with respect to the vertical direction with increasing depth. This is different from the classical prediction in homogeneous materials. The presence of the nonlocal scale effect can also change the geometry property of displacement fields. In addition, the amplitude of the vertical displacement is attenuated along the depth, while the increment of the nonlocal parameters can strengthen the attenuation.

Finite-difference modeling and characteristics analysis of Rayleigh waves in anisotropic-viscoelastic media

Effects of anisotropy and viscoelasticity of Earth media on Rayleigh waves propagation are studied by numerical simulation based on a finite-difference (FD) scheme, which calculates the spatial derivatives via a 12th-order staggered grid difference operator and the time derivatives through the fourth-order Runge-Kutta method. In the homogeneous half-space, the accuracy of FD algorithm is tested against the isotropic-elastic (IE) analytical solution; the calculation method of the theoretical phase velocities of the anisotropic-viscoelastic (AV) Rayleigh waves is developed to verify the correctness of modeling results; the characteristics of Rayleigh-wave are analyzed by comparisons among IE, anisotropic-elastic (AE) and AV modeling results in terms of the wave field snapshots, the synthetic seismograms, and the dispersive images, respectively. Then, the two-layer models and the four-layer models are utilized to further analyze the characteristics of Rayleigh waves in the AV layered media. Results show the substantial differences among these three kinds of media. Anisotropy of medium leads to the significant changes of Rayleigh-wave in amplitude, waveform and phase velocity due to anisotropy of velocity, but does not cause the phase velocity dispersion of Rayleigh-wave. Viscoelasticity of media arouses the amplitude attenuation and phase velocity dispersion of Rayleigh-wave. This is the first report for FD modeling and characteristics analysis of Rayleigh-wave in the AV media, which will provide a valuable reference for near-surface geophysical investigation.

Modeling linear Rayleigh wave sound fields generated by angle beam wedge transducers

In this study, the reciprocity theorem for elastodynamics is transformed into integral representations, and the fundamental solutions of wave motion equations are obtained using Green’s function method that yields the integral expressions of sound beams of both bulk and Rayleigh waves. In addition to this, a novel surface integral expression for propagating Rayleigh waves generated by angle beam wedge transducers along the surface is developed. Simulation results show that the magnitudes of Rayleigh wave displacements predicted by this model are not dependent on the frequencies and sizes of transducers. Moreover, they are more numerically stable than those obtained by the 3-D Rayleigh wave model. This model is also applicable to calculation of Rayleigh wave beams under the wedge when sound sources are assumed to radiate waves in the forward direction. Because the proposed model takes into account the actual calculated sound sources under the wedge, it can be applied to Rayleigh wave transducers with different wedge geometries. This work provides an effective and general tool to calculate linear Rayleigh sound fields generated by angle beam wedge transducers.

Quantifying tissue viscoelasticity using optical coherence elastography and the Rayleigh wave model.

This study demonstrates the feasibility of using the Rayleigh wave model (RWM) in combination with optical coherence elastography (OCE) technique to assess the viscoelasticity of soft tissues. Dispersion curves calculated from the spectral decomposition of OCE-measured air-pulse induced elastic waves were used to quantify the viscoelasticity of samples using the RWM. Validation studies were first conducted on 10% gelatin phantoms with different concentrations of oil. The results showed that the oil increased the viscosity of the gelatin phantom samples. This method was then used to quantify the viscoelasticity of chicken liver. The Young’s modulus of the chicken liver tissues was estimated as E=2.04±0.88??kPa with a shear viscosity ?=1.20±0.13??Pa?s. The analytical solution of the RWM correlated very well with the OCE-measured phased velocities (R2=0.96±0.04). The results show that the combination of the RWM and OCE is a promising method for noninvasively quantifying the biomechanical properties of soft tissues and may be a useful tool for detecting disease.

Modeling of atmospheric-coupled Rayleigh waves on planets with atmosphere: From Earth observation to Mars and Venus perspectives.

Acoustic coupling between solid Earth and atmosphere has been observed since the 1960s, first from ground-based seismic, pressure, and ionospheric sensors and since 20 years with various satellite measurements, including with global positioning system (GPS) satellites. This coupling leads to the excitation of the Rayleigh surface waves by local atmospheric sources such as large natural explosions from volcanoes, meteor atmospheric air-bursts, or artificial explosions. It contributes also in the continuous excitation of Rayleigh waves and associated normal modes by atmospheric winds and pressure fluctuations. The same coupling allows the observation of Rayleigh waves in the thermosphere most of the time through ionospheric monitoring with Doppler sounders or GPS. The authors review briefly in this paper observations made on Earth and describe the general frame of the theory enabling the computation of Rayleigh waves for models of telluric planets with atmosphere. The authors then focus on Mars and Venus and give in both cases the atmospheric properties of the Rayleigh normal modes and associated surface waves compared to Earth. The authors then conclude on the observation perspectives especially for Rayleigh waves excited by atmospheric sources on Mars and for remote ionospheric observations of Rayleigh waves excited by quakes on Venus.

Multi-Gaussian beam model for ultrasonic surface waves with angle beam rectangular transducers

Rayleigh waves propagating in an elastic surface are commonly used for the near surface flaw detection and material characterization. However, unlike the bulk wave case, there are seldom three-dimensional models to be provided for the Rayleigh waves. In the past decade, multi-Gaussian beam models have been gradually developed and perfectly applied to solve many complicated propagation problems of bulk waves. However, up to date they have not been extended to the simulation of the Rayleigh waves. By combining the Rayleigh wave Green function and the multi-Gaussian beam model, a three-dimensional Rayleigh wave model is presented to calculate the beam fields radiated from a rectangular transducer mounted on the Lucite wedge. Furthermore, some simulation results of the provided method are compared to those of a more exacted point source model. It is shown that the multi-Gaussian surface wave model has good capability in both computational accuracy and efficiency.

Low dispersive modeling of Rayleigh waves on partly staggered grids

In elastic media, finite-difference (FD) implementations of free-surface (FS) boundary conditions on partly staggered grid (PSG) use the highly dispersive vacuum formulation (VPSG). The FS boundary is embedded into a “vacuum” grid layer (null Lame’s constants and negligible density values) where the discretized equations of motion allow computing surface displacements. We place a new set of compound (stress-displacement) nodes along a planar FS and use unilateral mimetic FD discretization of the zero-traction conditions for displacement computation (MPSG). At interior nodes, MPSG reduces to standard VPSG methods and applies fourth-order centered FD along cell diagonals for staggered differentiation combined with nodal second-order FD in time. We perform a dispersion analysis of these methods on a Lamb’s problem and estimate dispersion curves from the phase difference of windowed numerical Rayleigh pulses at two FS receivers. For a given grid sampling criterion (e.g., six or ten nodes per reference S wavelength λ S), MPSG dispersion errors are only a quarter of the VPSG method. We also quantify root-mean-square (RMS) misfits of numerical time series relative to analytical waveforms. MPSG RMS misfits barely exceed 10 % when nine nodes sample the minimum S wavelength \(\lambda _{\text {MIN}}^{\mathrm {S}}\) in transit (along distances \(\sim \)145\(\lambda _{\text {MIN}}^{\mathrm {S}}\)). In same tests, VPSG RMS misfits exceed 70 %. We additionally compare MPSG to a consistently fourth-order mimetic method designed on a standard staggered grid. The latter equates the former’s dispersion errors on grids twice denser and shows higher RMS precision only on grids with six or less nodes per \(\lambda _{\text {MIN}}^{\mathrm {S}}\).

Teleseismic magnetic effects (TMDs) of 2011 Tohoku earthquake

Anomalous magnetic variations were observed by ground magnetometers in East Asia area after the 2011 Tohoku earthquake. Some earlier reports showed that the seismo‐magnetic variations have obvious amplitude around the epicenter, we emphasis here that the variations can still be notable at stations 2000–4000 km away from epicenter, and we define it as teleseismic magnetic disturbances (TMDs). TMDs appear about 8 min later after the arrival of seismic Rayleigh waves at teleseismic distances and propagate at a horizontal velocity of 3.9 ± 0.1 km/s. The wave‐like TMDs last for no longer than 10 min and have a main period of 2.1–3.3 min. TMDs are not generated by direct effects of processes in focal area crust or tsunami waves, instead, their properties consist with the Rayleigh wave model of seismo‐ionospheric disturbances. Hence, we conclude that the TMDs are the magnetic manifestation of seismotraveling ionospheric disturbances (STIDs) generated by the interaction between the ionosphere and atmosphere through acoustic waves launched by traveling Rayleigh waves. Our findings contribute to the knowledge of seismo‐electromagnetic effects in the atmosphere‐ionosphere system and further our understanding of couplings between various spheres of the Earth.

表面波伝播シミュレーションにおける粒子法の精度評価

表面波伝播シミュレーションにおける粒子法の精度評価

Постановка смешанных задач для упругой полуплоскости в рамках асимптотической модели для волны Рэлея

Постановка смешанных задач для упругой полуплоскости в рамках асимптотической модели для волны Рэлея

EXTENSION OF THE PUFEM TO ELASTIC WAVE PROPAGATION IN LAYERED MEDIA

This work deals with the extension of the partition of unity finite element method (PUFEM) "(Comput. Meth. Appl. Mech. Eng.139 (1996) pp. 289–314; Int. J. Numer. Math. Eng.40 (1997) 727–758)" to solve wave problems involving propagation, transmission and reflection in layered elastic media. The proposed method consists of applying the plane wave basis decomposition to the elastic wave equation in each layer of the elastic medium and then enforce necessary continuity conditions at the interfaces through the use of Lagrange multipliers. The accuracy and effectiveness of the proposed technique is determined by comparing results for selected problems with known analytical solutions. Complementary results dealing with the modeling of pure Rayleigh waves are also presented where the PUFEM model incorporates information about the pressure and shear waves rather than the Rayleigh wave itself.

An improved vacuum formulation for 2D finite-difference modeling of Rayleigh waves including surface topography and internal discontinuities

Rayleigh waves are generated along the free surface and their propagation can be strongly influenced by surface topography. Modeling of Rayleigh waves in the near surface in the presence of topography is fundamental to the study of surface waves in environmental and engineering geophysics. For simulation of Rayleigh waves, the traction-free boundary condition needs to be satisfied on the free surface. A vacuum formulation naturally incorporates surface topography in finite-difference (FD) modeling by treating the surface grid nodes as the internal grid nodes. However, the conventional vacuum formulation does not completely fulfill the free-surface boundary condition and becomes unstable for modeling using high-order FD operators. We developed a stable vacuum formulation that fully satisfies the free-surface boundary condition by choosing an appropriate combination of the staggered-grid form and a parameter-averaging scheme. The elastic parameters on the topographic free surface are updated with exactly the same treatment as internal grid nodes. The improved vacuum formulation can accurately and stably simulate Rayleigh waves along the topographic surface for homogeneous and heterogeneous elastic models with high Poisson’s ratios ([Formula: see text]). This method requires fewer grid points per wavelength than the stress-image-based methods. Internal discontinuities in a model can be handled without modification of the algorithm. Only minor changes are required to implement the improved vacuum formulation in existing 2D FD modeling codes.

Pseudospectral modeling and dispersion analysis of Rayleigh waves in viscoelastic media

Multichannel Analysis of Surface Waves (MASW) is one of the most widely used techniques in environmental and engineering geophysics to determine shear-wave velocities and dynamic properties, which is based on the elastic layered system theory. Wave propagation in the Earth, however, has been recognized as viscoelastic and the propagation of Rayleigh waves presents substantial differences in viscoelastic media as compared with elastic media. Therefore, it is necessary to carry out numerical simulation and dispersion analysis of Rayleigh waves in viscoelastic media to better understand Rayleigh-wave behaviors in the real world. We apply a pseudospectral method to the calculation of the spatial derivatives using a Chebyshev difference operator in the vertical direction and a Fourier difference operator in the horizontal direction based on the velocity–stress elastodynamic equations and relations of linear viscoelastic solids. This approach stretches the spatial discrete grid to have a minimum grid size near the free surface so that high accuracy and resolution are achieved at the free surface, which allows an effective incorporation of the free surface boundary conditions since the Chebyshev method is nonperiodic. We first use an elastic homogeneous half-space model to demonstrate the accuracy of the pseudospectral method comparing with the analytical solution, and verify the correctness of the numerical modeling results for a viscoelastic half-space comparing the phase velocities of Rayleigh wave between the theoretical values and the dispersive image generated by high-resolution linear Radon transform. We then simulate three types of two-layer models to analyze dispersive-energy characteristics for near-surface applications. Results demonstrate that the phase velocity of Rayleigh waves in viscoelastic media is relatively higher than in elastic media and the fundamental mode increases by 10–16% when the frequency is above 10 Hz due to the velocity dispersion of P and S waves.