Quasi-shear Waves Research Articles

Angular and Frequency-Dependent Wave Velocity and Attenuation in Fractured Porous Media

Wave-induced fluid flow generates a dominant attenuation mechanism in porous media. It consists of energy loss due to P-wave conversion to Biot (diffusive) modes at mesoscopic-scale inhomogeneities. Fractured poroelastic media show significant attenuation and velocity dispersion due to this mechanism. The theory has first been developed for the symmetry axis of the equivalent transversely isotropic (TI) medium corresponding to a poroelastic medium containing planar fractures. In this work, we consider the theory for all propagation angles by obtaining the five complex and frequency-dependent stiffnesses of the equivalent TI medium as a function of frequency. We assume that the flow direction is perpendicular to the layering plane and is independent of the loading direction. As a consequence, the behaviour of the medium can be described by a single relaxation function. We first consider the limiting case of an open (highly permeable) fracture of negligible thickness. We then compute the associated wave velocities and quality factors as a function of the propagation direction (phase and ray angles) and frequency. The location of the relaxation peak depends on the distance between fractures (the mesoscopic distance), viscosity, permeability and fractures compliances. The flow induced by wave propagation affects the quasi-shear (qS) wave with levels of attenuation similar to those of the quasi-compressional (qP) wave. On the other hand, a general fracture can be modeled as a sequence of poroelastic layers, where one of the layers is very thin. Modeling fractures of different thickness filled with CO2 embedded in a background medium saturated with a stiffer fluid also shows considerable attenuation and velocity dispersion. If the fracture and background frames are the same, the equivalent medium is isotropic, but strong wave anisotropy occurs in the case of a frameless and highly permeable fracture material, for instance a suspension of solid particles in the fluid.

Reection and refraction of plane waves at the interface between magnetoelectroelastic and liquid media

This paper analyzes the reflection and refraction of plane wave incidences at the interface between magnetoelectroelastic (MEE) and liquid media. The MEE medium is assumed to be transversely isotropic and the liquid medium to be nonviscous. Three cases, i.e., the coupled quasipressure wave incidence from the MEE medium, the coupled quasi-shear vertical wave incidence from the MEE medium, and the pressure wave incidence from the liquid medium, are discussed. The expressions of reflection and transmission coefficients varying with the incident angle are obtained. This investigation would be useful to the MEE acoustic device field.

Acousto-optic study of the unusual cases of reflection of bulk elastic waves in a paratellurite crystal

The propagation and reflection of bulk ultrasonic waves in a paratellurite crystal have been investigated by the acousto-optic method. Simultaneous excitation of two acoustic waves by one piezoelectric transducer has been observed in a cell of complex, specially chosen configuration. Maximally efficient conversion of the energy of a quasi-longitudinal wave into a single reflected quasi-shear wave with a walkoff angle 72° is implemented for the reflection from a free crystal boundary. The wave beam compression by a factor of 7 is observed for this reflection. A reflection (close to backward) of waves with an angle of spatial separation of the incident and reflected beams equal to 8° is also implemented. The acousto-optic figure of merit values of the medium are calculated for arbitrary propagation and polarization directions of the interacting light and ultrasonic waves. Possible applications of the effects studied in acousto-optics and acoustoelectronics are indicated.

Distinguishing between stress-induced and structural anisotropy at Mount Ruapehu volcano, New Zealand

[1] We have created a benchmark of spatial variations in shear wave anisotropy around Mount Ruapehu, New Zealand, against which to measure future temporal changes. Anisotropy in the crust is often assumed to be caused by stress-aligned microcracks, and the polarization of the fast quasi-shear wave (ϕ) is thus interpreted to indicate the direction of maximum horizontal stress, but can also be due to aligned minerals or macroscopic fractures. Changes in seismic anisotropy have been observed following a major eruption in 1995/96 and were attributed to changes in stress from the depressurization of the magmatic system. Three-component broadband seismometers have been deployed to complement the permanent stations that surround Ruapehu, creating a combined network of 34 three-component seismometers. This denser observational network improves the resolution with which spatial variations in seismic anisotropy can be examined. Using an automated shear wave splitting analysis, we examine local earthquakes in 2008. We observe a strong azimuthal dependence of ϕ and so introduce a spatial averaging technique and two-dimensional tomography of recorded delay times. The anisotropy can be divided into regions in which ϕ agrees with stress estimations from focal mechanism inversions, suggesting stress-induced anisotropy, and those in which ϕ is aligned with structural features such as faults, suggesting structural anisotropy. The pattern of anisotropy that is inferred to be stress related cannot be modeled adequately using Coulomb modeling with a dike-like inflation source. We suggest that the stress-induced anisotropy is affected by loading of the volcano and a lithospheric discontinuity.

Propagation of quasishear waves in prestressed materials with unbonded layers

The propagation of quasi-shear waves along the layers of a prestressed composite material is described using the three-dimensional linearized theory of elasticity. The composite consists of two alternating layers that are free to slip relative each other. The dispersion equation is analyzed numerically. The effect of the prestresses on the wave velocity is studied

Evolution of the fronts of quasi-compressional and quasi-shear discontinuous waves in inhomogeneous transversely isotropic elastic media

The propagation and evolution of the fronts of discontinuous waves in inhomogeneous transversely isotropic elastic media are studied. A method to draw evolving rays and fronts is proposed. Geometrical singularities on the fronts are studied for different parameters of anisotropy and inhomogeneity

Wavenumber Sampling Issues in 2.5D Frequency Domain Seismic Modelling

There are several important wavenumber sampling issues associated with 2.5D seismic modelling in the frequency domain, which need careful attention if accurate results are to be obtained. At certain critical wavenumbers there exist rapid disruptions in the mainly smooth oscillatory spectra. The amplitudes of these disruptions can be very large, and this affects the accuracy of the inverse Fourier transformed frequency-space domain solution. In anisotropic elastic media there are critical wavenumbers associated with each wave mode—the quasi-P (qP) wave, and the two quasi-shear (qS1 and qS2) waves. A small wavenumber sampling interval is desirable in order to capture the highly oscillatory nature of the wavenumber spectrum, especially at increasing distance from the source. Obviously a small wavenumber sampling interval adds greatly to the computational effort because a 2D problem must be solved for every wavenumber and every frequency. The discretisation should be carried out up to some maximum wavenumber, beyond which the field becomes evanescent (exponentially decaying or diffusive). For receivers close to the source, activity persists beyond the critical wavenumber associated with the minimum shear wave velocity in the model. Fortunately, for receivers well removed from the source, the contribution from the evanescent energy is negligible and so there is no need to sample beyond this critical wavenumber. Sampling at Gauss–Legendre spacings is a satisfactory approach for acoustic media, but it is not practical in elastic media due to the difficulty of partitioning the integration around the different critical wavenumbers. We found to our surprise that in transversely isotropic media, the critical wavenumbers are independent of wave direction, but always occur at those wavenumbers corresponding to the maximum phase velocities of the three wave modes (qP, qS1 and qS2), which depend only on the elastic constants and the density. Additionally, we have observed that intermediate layers between source and receiver can filter out to a large degree, the sharp irregularities around the critical wavenumbers in the ω–k y spectra. We have found that, using the spectral element method, the singularities (poles) at the critical wavenumbers which exist with analytic solutions, do not arise. However, the troublesome spike-like behaviour still occurs and can be damped out without distorting the spectrum elsewhere, through the introduction of slight attenuation.

Caustics in a periodically layered transversely isotropic medium with vertical symmetry axis

Wave propagation in a finely layered medium is a very important topic in seismic modelling and inversion. Here we analyse non-vertical wave propagation in a periodically layered transversely isotropic (VTI) medium and show that the evanescent (attenuation) zones in the frequency-horizontal slowness domain result in caustics in the group velocity domain. These caustics, which may appear for both the quasi-compressional (qP) and quasi-shear (qSV) wave surfaces are frequency dependent but display weak dependence at low frequencies. The caustics computed for a specific frequency differ from those observed at the low- and high-frequency limits. We illustrate these caustics with a few numerical examples and snapshots computed for both qP- and qSV-wave types.

Determination of the minimum entropy configuration and velocity surfaces for elastic–plastic wave scattering at grain boundaries

A recent model based on full elastic anisotropy and crystal plasticity predicted the existence of multiple wave configurations during the interaction of stress waves with grain boundaries. Since the multiple wave configuration scenario cannot exist in nature, the principle of minimum entropy production is applied in the current work to find the most probable configuration. A large amplitude transmitted quasi-longitudinal wave is predicted for the given bicrystal orientation studied due to the wave propagating near a [001] direction and thus requiring large stress given the very low Schmid factor in this direction (for nickel aluminide (NiAl) as a model material). Anisotropic elastic–plastic velocity surfaces for quasi-longitudinal and quasi-shear waves in NiAl have also been constructed to gain an understanding of the general nature of plastic waves as a function of crystallographic direction.

Quasi-shear wave ray tracing by wavefront construction in 3-D, anisotropic media

Wavefront construction (WFC) methods provide robust tools for computing ray theoretical traveltimes and amplitudes for multivalued wavefields. They simulate a wavefront propagating through a model using a mesh that is refined adaptively to ensure accuracy as rays diverge during propagation. However, an implementation for quasi-shear (qS) waves in anisotropic media can be very difficult, since the two qS slowness surfaces and wavefronts often intersect at shear-wave singularities. This complicates the task of creating the initial wavefront meshes, as a particular wavefront will be the faster qS-wave in some directions, but slower in others. Analogous problems arise during interpolation as the wavefront propagates, when an existing mesh cell that crosses a singularity on the wavefront is subdivided. Particle motion vectors provide the key information for correctly generating and interpolating wavefront meshes, as they will normally change slowly along a wavefront. Our implementation tests particle motion vectors to ensure correct initialization and propagation of the mesh for the chosen wave type and to confirm that the vectors change gradually along the wavefront. With this approach, the method provides a robust and efficient algorithm for modeling shear-wave propagation in a 3-D, anisotropic medium. We have successfully tested the qS-wave WFC in transversely isotropic models that include line singularities and kiss singularities. Results from a VTI model with a strong vertical gradient in velocity also show the accuracy of the implementation. In addition, we demonstrate that the WFC method can model a wavefront with a triplication caused by intrinsic anisotropy and that its multivalued traveltimes are mapped accurately. Finally, qS-wave synthetic seismograms are validated against an independent, full-waveform solution.

Wave propagation in transversely isotropic porous piezoelectric materials

Wave propagation in porous piezoelectric material (PPM), having crystal symmetry 6 mm, is studied analytically. Christoffel equation is derived for the propagation of plane harmonic waves in such a medium. The roots of this equation give four complex wave velocities which can propagate in such materials. The phase velocities of propagation and the attenuation quality factors of all these waves are described in terms of complex wave velocities. Phase velocities and attenuation of the waves in PPM depend on the phase direction. Numerical results are computed for the PPM BaTiO 3. The variation of phase velocity and attenuation quality factor with phase direction, porosity and the wave frequency is studied. The effects of anisotropy and piezoelectric coupling are also studied. The phase velocities of two quasi dilatational waves and one quasi shear waves get affected due to piezoelectric coupling while that of type 2 quasi shear wave remain unaffected. The phase velocities of all the four waves show non-dispersive behavior after certain critical high frequency. The phase velocity of all waves decreases with porosity while attenuation of respective waves increases with porosity of the medium. The characteristic curves, including slowness curves, velocity curves, and the attenuation curves, are also studied in this paper.

Estimation of Intrinsic Damage State in Materials using Non-local Perturbation: Application to Active Health Monitoring

Quantifying damage parameters from macroscale wave signals has gained enormous popularity in recent years in the field of structural health monitoring (SHM). However, suitable parametric variation which manifests such lower scale effect to predict incubation of damage is an intricate subject. Hence, a relatively simpler way of quantifying the damage state is discussed in this article. An effort has been made to preset the dispersive solution of Cristoffel equation using non-local formulation. The solution of non-local Cristoffel equation is presented for isotropic (iron crystal—metal) and anisotropic material (cubic crystal—gallium arsenide—semiconductor). Conventional solution of Cristoffel equation is non-dispersive, but this study shows that the wave modes such as quasi-longitudinal, quasi-shear wave speeds are dispersive when non-local theory is employed. However, velocity directions are not perturbed from the non-local Cristoffel equation. Non-local parameters can be used as a migratory parameter for elastodynamic analysis at the macroscale considering the intrinsic length scale effect. The variations of wave velocities (phase velocities) in frequency domain are also presented under the influence of non-local parameter. This study has two implications in the SHM field. The dispersive nature of waves in a periodic crystal structure can be obtained from a MD study of the undamaged condition. Hence, a suitable value of the non-local parameter can be found and can be used as an input parameter in the derived equation in this article. On the other hand, the experimental study on wave velocity can be used to estimate the suitable non-local parameter as a damage parameter which carries the property of damaged crystals. In this article, the later approach has been adopted to study using COMSOL multiphysics software. Three different types of damages are considered inside the materials mentioned above. The effect on wave velocity is found and from the reverse analysis, non-local parameter is determined.

Nonlinear traveltime inversion scheme for crosshole seismic tomography in tilted transversely isotropic media

Crosshole seismic tomography often is applied to image the velocity structure of an interwell medium. If the rocks are anisotropic, the tomographic technique must be adapted to the complex situation; otherwise, it leads to a false interpretation. We propose a nonlinear kinematic inversion method for crosshole seismic tomography in composite transversely isotropic media with known dipping symmetry axes. This method is based on a new version of the first-order traveltime perturbation equation. It directly uses the derivative of the phase velocity rather than the eigenvectors of the body-wave modes to overcome the singularity problem for application to the two quasi-shear waves. We applied an iterative nonlinear solver incorporating our kinematic ray-tracing scheme and directly compute the Jacobian matrix in an arbitrary reference medium. This reconstructs the five elastic moduli or Thomsen parameters from the first-arrival traveltimes of the three seismic body waves (qP, qSV, qSH) in strongly and weakly anisotropic media. We conducted three synthetic experiments that involve determining anisotropic parameters for a homogeneous rock, reconstructing a fault embedded in a strongly anisotropic background, and imaging a complicated four-layer model containing a small channel and a buried dipping interface. We compared results of our nonlinear inversion method with isotropic tomography and the traditional linear anisotropic inversion scheme, which showed the capability and superiority of the new scheme for crosshole tomographic imaging.

Velocity sensitivity of seismic body waves to the anisotropic parameters of a TTI-medium

We formulate the derivatives of the phase and group velocities for each of the anisotropic parameters in a tilted transversely isotropic medium (TTI-medium). This is a common geological model in seismic exploration and has five elastic moduli or related Thomsen parameters and two orientation angles defining the axis of symmetry of the rock. We present two independent methods to compute the derivatives and examine the formulae with real anisotropic rocks. The formulations and numerical computations do not encounter any singularity problem when applied to the two quasi shear waves, which is a problem with other approaches. The two methods yield the same results, which show in a quantitative way the sensitivity behaviour of the phase and the group velocities to all of the elastic moduli or Thomsen's anisotropic parameters as well as the orientation angles in the 2D and 3D cases. One can recognize the dominant (strong effect) and weak (or 'dummy') parameters for the three seismic body-wave modes (qP, qSV, qSH) and their effective domains over the whole range of phase-slowness directions. These sensitivity patterns indicate the possibility of nonlinear kinematic inversion with the three wave modes for determining the anisotropic parameters and imaging an anisotropic medium.

Propagation and interaction of weakly nonlinear elastic plane waves in a cubic crystal

Using the asymptotic method of weakly nonlinear acoustics we analyze the propagation and interaction of elastic plane waves in a cubic crystal. We study the case of an arbitrary direction in a cube face (0 0 1) as well as three selected directions in particular the direction along the cube diagonal which is the threefold symmetric axis of a crystal. In the latter case new evolution equations derived in [W. Domański, Asymptotic equations for weakly nonlinear elastic waves in a cubic crystal, Int. Ser. Num. Math. 129 (1999) 233–241; W. Domański, Weakly nonlinear elastic plane waves in a cubic crystal, Contemp. Math. 255 (2000) 45–61] are recalled. These equations (called complex Burgers equations) describe a quadratically nonlinear interaction of collinear shear and quasi-shear elastic waves in a cubic crystal. Tables of interaction coefficients are displayed in the case of an arbitrary direction in a cube face (0 0 1) revealing possible quadratically nonlinear interactions between all waves in that case.

The traveltime perturbations for seismic body waves in factorized anisotropic inhomogeneous media

SUMMARY The traveltime perturbation equations for the quasi-compressional and the two quasi-shear waves propagating in a factorized anisotropic inhomogeneous (FAI) media are derived. The concept of FA1 media simplifies considerably these equations. In the FA1 medium, the density normalized elastic parameters uZlkl(x,) can be described by the relation urlkl(x,) = f2(x,)A,,kl, where are constants, independent of coordinates x,, and f2(xz) is a continuous smooth function of x,. The types of anisotropy (A,,kl) and inhomogeneity [f(x,)] are not restricted. The traveltime perturbations of individual seismic body waves (qP, qS1 and 4.92) propagating in the FA1 medium depend, of course, both on the structural pertubations [6f2(xz)J and on the anisotropy perturbations (6A+/), but both these effects are fully separated. The perturbation equations for the time delay between the two @-waves propagating in the FA1 medium are simplified even more. If the unperturbed (background) medium is isotropic, the perturbation of the time delay does not depend on the structural perturbations 6f2(x,) at all. This striking result, valid of course only in the framework of first-order perturbation theory, will simplify considerably the interpretation of the time delay between the two split @-waves in inhomogeneous anisotropic media. Numerical examples are presented.

Wave propagation in an incompressible transversely isotropic fibre-reinforced elastic media

The boundary conditions at free surface of an incompressible, transversely isotropic elastic half-space are satisfied to obtain the reflection coefficients for the case when outer slowness section is re-entrant. Two quasi-shear waves will be reflected for an angular range of direction of incident wave. The numerical illustrations of reflection coefficients are presented graphically for three arbitrary materials.

Vector imaging of converted wave data in laterally uniform media with VTI anisotropy

A vector imaging method has been developed for PS-converted waves in laterally homogeneous vertically transverse isotropic (VTI) media. It decomposes the converted-wave data into two upgoing quasi-shear waves ([Formula: see text] and [Formula: see text]) within the prestack migration algorithm according to subsurface image and surface receiver locations. Because the decomposition is performed as part of the migration, it is consistent with the dip and polarization of the seismic events, unlike traditional algorithms that use premigration rotations. Two shear-wave images with potentially enhanced resolution are formed simultaneously from the vector migration. The effects of VTI anisotropy on PS-converted wave imaging and the capability of the PS vector imaging algorithm to provide enhanced images are illustrated using a point-scatterer model.

Bulk conical and surface helical acoustic waves in transversely isotropic cylinders; application to the stiffness tensor measurement

A point-source-point-receiver technique, based on laser generation and laser detection of acoustic waves, allows determination of mechanical properties of anisotropic cylinders. The anisotropic nature of the material and the geometry of the samples make the acoustic signature difficult to interpret. In addition to multiple surface waves, quasi-longitudinal and quasi-shear bulk waves are diffracted and acoustic rays are reflected with or without mode conversion at the cylinder surface. Moreover both bulk and surface diffracted waves have a dispersive behavior. To bypass the intricacies, wave fronts are synthesized with signals provided by scanning a straight line on the cylinder with the laser point source. Conical waves propagating in the volume and helical waves propagating along the surface are then numerically produced. The recovery of the stiffness-tensor components is based on the inversion of the bulk waves, phase velocities. The method is presented and applied to signals simulated or experimentally recorded for a composite material. The five independent stiffness coefficients of the hexagonal symmetry are thus measured with waveforms provided by a single scan along the cylinder surface. The method provides a unique mean for the noncontact measurement of elastic properties of cylindrical parts.

Nonlinear anisotropic description for the thermomechanical response of shocked single crystals: Inelastic deformation

A nonlinear anisotropic continuum framework for describing the thermoelastic-plastic response of single crystals shocked along arbitrary orientations has been developed. Our modeling approach incorporates nonlinear elasticity, crystal plasticity, and thermodynamic consistency within an incremental tensor formulation. Crystal plasticity was incorporated by considering dislocation motion along specified slip planes. The theoretical developments presented here are sufficiently general to also accommodate other types of inelastic deformation mechanisms. As representative applications of the theoretical developments, numerical simulations of shock wave propagation in lithium fluoride (LiF) and copper single crystals are presented and compared to wave profile data for several crystal orientations. Simulations of shock wave propagation along low-symmetry directions, where data are not available, are also presented to examine the propagation of quasilongitudinal and quasishear waves in crystals undergoing elastic-plastic deformation. Temperature calculations for the shocked single crystals are discussed.