Phase Velocity Surfaces Research Articles

Behavior of S waves in vicinity of singularity point in elliptic orthorhombic media

In low-symmetry anisotropic models, the S waves are generally coupled and have very complex behavior due to triplications and singularity points. The presence of singularity point affects S-wave phase and group velocity surface in a simple elliptical orthorhombic model. This model has no triplications but has a singularity point in one of the symmetry planes. We find that the slowness ellipse (defined by the zero Gaussian curvature) is the image of a singularity point on the group velocity surface, and the group velocity ellipse (also known as the internal refraction cone) is the image of a singularity point on the slowness surface. The equations for the slowness and group velocity ellipses are explicitly defined. The square of S1- and S2-wave lacunae also is computed.

Uncertainty quantification of phase velocity surface waves multy-modal inversion using machine learning

The paper is devoted to uncertainty quantification of the inverse problem solution of the multichannel analysis of surface waves method - the inversion of the curves of the phase velocity via frequency dependence. The uncertainty estimation approach is based on the Monte Carlo sampling strategy and a multilayer fully connected artificial neural network to approximate nonlinear dependence of shear wave velocity and layers thickness via values of phase velocity surface waves. Frequency-dependent noise in the data and errors of the inverse operator are projected onto the inverse problem solution. The results of unimodal and multimodal inversion are compared on the example of synthetic data processing. The experimental results show that using of machine learning approaches makes it possible to quickly and accurately estimate the posterior probability density of the reconstructed velocity model parameters.

Eikonal surface wave tomography with smoothing splines—application to Southern California

SUMMARY The densification of both permanent and temporary seismic networks has raised new interest in surface wave eikonal tomography from which phase velocity maps can be obtained without resolving a tomographic inverse problem. However, eikonal tomography requires to reconstruct traveltime surfaces from a discrete number of measurements obtained at the station locations, which can be challenging. We present a new method to reconstruct these traveltime surfaces with smoothing splines discretized in a regular 2-D Cartesian grid. We impose Neumann boundary conditions so that the phase gradients on the edges of the grid are equal to the apparent slownesses of the average plane wave along the normal direction measured by beamforming. Using the eikonal equation, phase velocity maps are then derived from the norm of the gradient of the interpolated traveltime maps. The method is applied to Rayleigh waves recorded by the Southern California Seismic Network to derive phase velocity surfaces. Robust, stable and finely resolved phase velocity maps at 25 and 33 s period are obtained after averaging the phase velocity maps derived from the analysis of a selection of recent large (Mw ≥ 6.5) teleseismic events. The phase velocity map at 25 s mainly constrains the thickness of the Southern California crust, with results that are in excellent agreement with previous tomographic studies.

Estimation of parameters and thickness of a horizontal layer of arbitrary anisotropy from P-wave moveout

We present an inversion scheme to estimate anisotropy parameters and thickness of a homogeneous layer of arbitrary anisotropy from traveltimes of a reflected P wave. The scheme is based on the weak-anisotropy approximation of P-wave moveout formula. For the description of anisotropy, we use the so-called A-parameters, which represent an alternative to standard elastic moduli specifying anisotropic media, and allow specification of anisotropy of any symmetry, orientation and strength. The proposed scheme allows estimation of, at most, nine of fifteen A-parameters, which describe P-wave propagation in the weak-anisotropy approximation. Therefore, generally, the reconstruction of the P-wave phase-velocity surface from the estimated parameters is impossible. If we deal, however, with higher-symmetry anisotropy (transversely isotropic, orthorhombic) and if information about the orientation of its symmetry elements is available, reconstruction of the phase-velocity surface is possible. Synthetic tests indicate that the inversion scheme works well even in situations, in which the use of other commonly used moveout formulae leads to unsatisfactory results.

Analysis of S waves singularity points in porous rock with two orthogonal sets of mesoscale fractures

SUMMARY The shear waves phase velocity surfaces in orthorhombic (ORT) and lower symmetry anisotropic models touch each other in one or more points resulting in so called singularity points or acoustic axes. These singularity points result in dramatic changes of velocities, amplitudes and polarizations creating problems in seismic data processing and analysis. Considering the frequency-dependent anisotropy due to mesoscale fractures in Chapman's model, we describe the singularity points in porous rock with two orthogonal sets of mesoscale fractures. First, we give the equations for frequency-dependent phase velocities of P, S1 and S2 waves in this anelastic ORT media. Then, we derive the expressions for frequency-dependent singularity points within the symmetry planes and discuss the conditions to detect the existence of singularity point. Finally, the influences of frequency, porosity, fracture density, fracture scale and saturating fluid style on the positions of singularity points within the symmetry plane are investigated.

Group velocity of light in biaxial crystals.

Based on the wave vector surface, we have derived the analytical expressions for the group velocity of light traveling in an optically biaxial crystal in an arbitrary direction. Our formulas are in terms of either the wave vector direction or the ray direction. An axis dispersion vector is introduced to specify the rotation of the dielectric frame in monoclinic and triclinic crystals, which is found to be responsible for the asymmetry of the group velocity surfaces of these two crystal systems when compared to their phase and ray velocity surfaces. Our formulas confirm that the group velocity of light in a nondissipative medium equals the ratio between the Poynting vector and the total energy density of the electromagnetic field. The group index of light is found to be decomposable into three parts: the ray index, the group index due to principal refractive index dispersion, and that due to axis dispersion. Numerical calculations are performed for the group velocity of light in ${\rm Sn}_2{\rm P}_2{\rm S}_6$ crystal at 550 nm wavelength. Our results show that the dispersion of the principal refractive indices has a considerable contribution to the group index.

Asymmetry of the group velocity of light in monoclinic crystals.

We have proved that compared to phase velocity and ray velocity surfaces, the group velocity surface of a monoclinic crystal has a reduced symmetry, due to the loss of the two mirror planes that contain the crystallographic b axis. We have derived a formula for calculation of the group velocity of the extraordinary light traveling in a principal plane of a biaxial crystal, which takes into account the rotation of the dielectric frame due to frequency dispersion. The maximum asymmetry of the group velocity of light traveling in the a-c plane is found to be 2.4% at 365 nm in BiB3O6 and 1.4% at 550 nm in Sn2P2S6.

Bulk Waves in the Infinite Electric-Magnetic-Elastic Plate with Mixed Boundary Conditions

A dynamic solution is presented for the propagation of waves in an electric-magneto-elastic plate composed of piezoelectric, piezomagnetic materials and elastic matrix. The electric-magneto-elastic plate is polarized along the thickness direction. The generalized displacements are expressed as the sum of the gradient of a scalar (dilatation wave) and the curl of a vector (shear wave). With the help of dynamic equilibrium equations and geometric equations, we can obtain dynamic equations of the dilatation wave and the shear wave. The conclusion that the types of the dilatation waves and the shear waves remain unchanged after being reflected by the boundary can be obtained through the analysis of these kinetic equations. The dispersion properties and phase velocity surface of the dilatation and shear wave can be obtained by solutions of dynamic equilibrium equations. Influences of the piezoelectric and piezomagnetic parameters on wave characteristics are discussed.

Geometrical characteristics of phase and group velocity surfaces in anisotropic media

ABSTRACTThe phase and group velocity surfaces are essential for wave propagation in anisotropic media. These surfaces have certain features that, especially, for shear waves result in complications for modelling and inversion of recorded wavefields. To analyse wave propagation in an anisotropic model, it is important to identify these features in both the phase and group domains. We propose few characteristics for this analysis: the energy flux angle, decomposed in the polar and azimuth angle correction angles and enhancement factor, which is able to characterize both singularity points and triplication zones. The very simple equation that controls the triplications is derived in the phase domain. The proposed characteristics are illustrated for elastic and acoustic anisotropic models of different symmetry classes.

Practical concept of traveltime inversion of simulated P-wave vertical seismic profile data in weak to moderate arbitrary anisotropy

We have developed a practical concept of compressional wave (P-wave) traveltime inversion in weakly to moderately anisotropic media of arbitrary symmetry and orientation. The concept provides sufficient freedom to explain and reproduce observed anisotropic seismic signatures to a high degree of accuracy. The key to this concept is the proposed P-wave anisotropy parameterization (A-parameters) that, together with the use of the weak-anisotropy approximation, leads to a significantly simplified theory. Here, as an example, we use a simple and transparent formula relating P-wave traveltimes to 15 P-wave A-parameters describing anisotropy of arbitrary symmetry. The formula is used in the inversion scheme, which does not require any a priori information about anisotropy symmetry and its orientation, and it is applicable to weak and moderate anisotropy. As the first step, we test applicability of the proposed scheme on a blind inversion of synthetic P-wave traveltimes generated in vertical seismic profile experiments in homogeneous models. Three models of varying anisotropy are used: tilted orthorhombic and triclinic models of moderate anisotropy (approximately 10%) and an orthorhombic model of strong anisotropy (>25%) with a horizontal plane of symmetry. In all cases, the inversion yields the complete set of 15 P-wave A-parameters, which make reconstruction of corresponding phase-velocity surfaces possible with high accuracy. The inversion scheme is robust with respect to noise and the source distribution pattern. Its quality depends on the angular illumination of the medium; we determine how the absence of nearly horizontal propagation directions affects inversion accuracy. The results of the inversion are applicable, for example, in migration or as a starting model for inversion methods, such as full-waveform inversion, if a model refinement is desired. A similar procedure could be designed for the inversion of S-wave traveltimes in anisotropic media of arbitrary symmetry.

Dispersion of a surface plasmon polaritons in a thin dielectric films surrounded by a two graphene layers

In this work, we investigated the features of the propagation of surface plasmon-polariton modes with a frequencies ω=(1014..1015)rad/s with a symmetric and antisymmetric distribution of the electric field in the planar structure of a «graphene-thin dielectric film-graphene». They are received the dispersion characteristics for these types of modes. Also we found a spectral range, where the dispersion waves, phase and group velocity surface waves are most sensitive to change of a chemical potential of a graphene layers.

Algebraic degree of a general group-velocity surface

Three algebraic surfaces — the slowness surface, the phase-velocity surface, and the group-velocity surface — play fundamental roles in the theory of seismic wave propagation in anisotropic elastic media. While the slowness (sometimes called phase-slowness) and phase-velocity surfaces are fairly simple and their main algebraic properties are well understood, the group-velocity surfaces are extremely complex; they are complex to the extent that even the algebraic degree, [Formula: see text], of a system of polynomials describing the general group-velocity surface is currently unknown, and only the upper bound of the degree [Formula: see text] is available. This paper establishes the exact degree [Formula: see text] of the general group-velocity surface along with two closely related to [Formula: see text] quantities: the maximum number, [Formula: see text], of body waves that may propagate along a ray direction in a homogeneous anisotropic elastic solid [Formula: see text] and the maximum number, [Formula: see text], of isolated, singularity-unrelated cusps of a group-velocity surface [Formula: see text].

Determination of parameters of viscoelastic anisotropy from ray velocity and ray attenuation: Theory and numerical modeling

We have developed and numerically tested a method for determining parameters of homogeneous viscoelastic anisotropy from measurements of wavefields generated by point sources. The method is based on complex algebra and consists of several steps. First, a complex energy velocity surface is constructed from the directionally dependent velocity and attenuation measured along a set of ray directions. Second, a complex slowness surface is computed using the relation of polar reciprocity between the energy velocity and slowness vectors. The energy velocity vectors are homogeneous, but the corresponding slowness vectors are inhomogeneous. Finally, the complex phase velocity surface is calculated and inverted using the Christoffel equation. The inversion is nonlinear and can be performed in iterations. Numerical tests for the P-wave in transversely isotropic media showed that the method performed well for a wide range of models covering strong as well as weak velocity anisotropy and various levels of attenuation. The method was compared with a simplified approximate inversion when the inhomogeneity of the complex slowness vector is neglected. The neglect of the slowness vector inhomogeneity results in a significantly lower accuracy of the retrieved attenuation parameters. Accuracy with errors less than 10% is achieved only if the attenuation anisotropy is weak. This condition is, however, strongly restrictive because attenuation anisotropy is usually significant being more pronounced than the velocity anisotropy for most of rocks.

Analysis of laser-generated ultrasonic force source at specimen surface and display of bulk wave in transversely isotropic plate by numerical method

Taking into account the effects of thermal diffusion and optical penetration, as well as the finite width and duration of the laser source, the laser-generated ultrasonic force source at surface vicinity is presented. The full acoustic fields of laser-generated ultrasonic bulk wave are obtained and displayed in transversely isotropic plate. The features of laser-generated ultrasound bulk waves are analyzed. The features of laser-generated ultrasonic bulk wave are in good agreement with the theoretical results (the phase velocity surfaces), demonstrating the validity of this simulation. The numerical results indicate that the features of laser-generated ultrasound waveforms in anisotropic specimen, different from the case in isotropic materials, have a close relation with the propagating plane and propagation direction. This method can provide insight to the generation and propagation of laser-generated ultrasonic bulk wave in transversely isotropic material.

Finsler metric and elastic constants for weak anisotropic media

Differential geometric expressions of elastic constants for a seismic ray path are studied based on Finsler geometry. A Finsler function named m th root metric is considered to discuss transverse isotropic media in weak anisotropic case. Finsler parameters in the m th root metric are estimated from phase velocity surfaces. The slight differences from an elliptic wavefront can be expressed by the Finsler parameters. It is found a correlation between the Finsler parameters and the weak anisotropy parameters consisted of elastic constants. Especially, a positivity of weak anisotropy parameter influences on a restriction of Finsler parameter. On the other hand, a geometric condition of Finsler parameter gives a limitation of weak anisotropy parameter. Moreover, the Berwald Gauss curvature of m th root metric induces a relationship between the spreading ray paths and the weak anisotropy parameter. Therefore, the seismic ray paths in weak isotropic media can be expressed by the Finslerian properties of m th root metric.

Investigation of Lamb elastic waves in anisotropic multilayered composites applying the Green’s matrix

This article presents a numerical study of dispersion characteristics of some symmetric and antisymmetric composites modelled as multilayered packets of layers with arbitrary anisotropy of each layer. The authors introduce a subsidiary boundary problem of three-dimensional elasticity theory for the system of partial differential equations describing the harmonic oscillations of the composite caused by a surface load. The problem reduces to a boundary problem for ordinary differential equations by employing the Fourier transform. An algorithm of constructing the Fourier transform of the Green's matrix of the given boundary problem is presented. The wave numbers of Lamb waves propagating in composites, their phase velocity surfaces and group wave surfaces are presented through the poles of the transform of the Green's matrix. The authors obtain the dispersion curves for different directions and frequencies and investigate the dispersion curves and surfaces of wave numbers, phase velocities and group wave surfaces for various composites. The numerical results are then compared with the results obtained by applying other methods.

The Inhomogeneous Waves in Pyroelectrics

In this paper, the general theory of inhomogeneous waves in pyroelectric medium is addressed. The difference from the homogeneous wave is that the wave propagation vector is not coincident with the attenuation vector. The attenuation angle, defined by the angle between wave propagation vector and attenuation vector, is found to be limited in the range of (−90°, 90°). It is found that increasing the attenuation angle will introduce more dissipation and anisotropy. In both L–S theory and inertial entropy theory, four wave modes are found in pyroelectric medium, which are temperature, quasitransverse I, II and quasilongitudinal. Although there is no independent wave mode for the electric field, it can still propagate with other wave modes. The variations of phase velocities and attenuations with propagation angle and attenuation angle are discussed. Phase velocity surfaces on anisotropic and isotropic planes are presented for different attenuation angle. It is found that attenuation angle almost doesn't influence the phase velocities of elastic waves in both anisotropic and isotropic planes. In contrast, the roles it plays on temperature wave are obvious. The effects of the positive and negative attenuation angles are not the same in anisotropic plane. In inertial theory, the results indicate that the temperature wave is almost the same with L–S theory, whereas the coefficient of the inertial entropy should be less than certain value.

Wave surfaces and wave velocities in optics of non-absorbing optically active media

General relations between phase normal vectors, refraction vectors and ray refraction vectors as well as between phase and ray velocities of plane waves are obtained for linear non-absorbing optically active (bianisotropic) media. For such media the geometric constraint on the wave surfaces is discussed. Equality of the group and ray velocities of the monochromatic plane waves in dispersive bianisotropic media is proven. It is shown that the degree of the ray surface equation generally exceeds the degree of the refraction surface and phase velocity surface equations but it cannot be greater than 108. In particular, for gyrotropic dielectric media of symmetry classes 3, , 4, , 4/m, 6, and 6/m it equals 12. The domain of applicability of the duality principle known in optics of anisotropic media is determined.

Propagation of harmonic plane waves in a general anisotropic porous solid

Wave propagation is studied in a general anisotropic poroelastic solid. The presence of dissipation due to fluid-viscosity as well as hydraulic anisotropy of pore permeability are also considered. Biot's theory is used to derive a system of modified Christoffel equations for the propagation of plane harmonic waves in porous media. A non-trivial solution of this system is ensured by a determinantal equation. This equation is separated into two different polynomial equations. One is the quartic equation whose roots represent the complex velocities of four attenuating waves in the medium. The other is a eighth-degree polynomial whose roots represent the vertical slowness values for the four waves propagating upward and downward in a finite porous medium. Procedure is explained to associate the numerically obtained roots with the waves propagating in the medium. The slowness surfaces of waves reflected at the boundary of the medium are computed for a realistic numerical model. The behaviours of phase velocity surfaces are analysed with the help of numerical examples.

Sensitivity analysis of an inverse procedure for determination of elastic coefficients for strong anisotropy

The elastic coefficients of anisotropic solids are often evaluated from measurements of phase or group velocities of ultrasonic bulk waves by the usage of inverse optimizing procedures. This paper discusses the effects of various factors on such procedures results for transversely isotropic solids with considerably strong anisotropy. First, the inverse determination of all elastic coefficients of unidirectional CFRP composite is briefly outlined. Then the results of the optimization are treated as exact values and the sensitivity of the optimizing process versus main considered sources of inaccuracies is analyzed. Results of extensive simulations are presented to illustrate the effect of input data distortion, input data incompleteness, and geometrical conversion from experimentally obtained group velocities into corresponding phase velocities used as input data for the optimizing procedure. The paper takes note of how information about the elastic coefficients can be extracted from the different segments of the phase velocity surface. The stability versus input data distortion for inversion from group velocities and phase velocities is compared and the importance of reliable geometrical converting from group into phase velocities is illustrated. An novel method for geometrical conversion of distorted group velocity data into corresponding phase velocities based on affine combinations of low-order polynomials is presented and compared with piecewise or high-order polynomial fitting.