Group Velocity Surfaces Research Articles

Shear-wave group-velocity surfaces in low-symmetry anisotropic media

Shear waves excited by natural sources constitute a significant part of useful energy recorded in downhole microseismic surveys. In rocks, such as fractured shales, exhibiting symmetries lower than transverse isotropy (TI), the shear wavefronts are always multivalued in certain directions, potentially complicating the data processing and analysis. This paper discusses a basic tool — the computation of the phase and group velocities of all waves propagating along a given ray — that intends to facilitate the understanding of geometries of the shear wavefronts in homogeneous anisotropic media. With this tool, arbitrarily complex group-velocity surfaces can be conveniently analyzed, providing insights into possible challenges to be faced when processing shear waves in anisotropic velocity models that have symmetries lower than TI. Among those challenges are complicated multipathing and the presence of cones of directions, known as internal refraction cones, in which no fast shear waves propagate and the entire shear portion of the body-wave seismic data consists of several branches of the slow shear wavefronts.

Bulk waves and dynamical behaviour in elastic solids reinforced by two families of strong fibres

In the study of wave propagation, bulk waves exist in infinite homogeneous bodies and propagate indefinitely without being interrupted by boundaries or interfaces. Such waves may be decomposed into finite plane waves propagating along arbitrary directions of the solid. The properties of these waves are determined by the dependence between the propagation direction and the constitutive properties of media. Three types of such waves may be distinguished in connection to the three displacement vectors which determine an acoustic polarization. These three polarization vectors are mutually orthogonal, but in most cases they are neither perpendicular nor parallel to the propagation direction. There are a number of papers and books describing wave front surfaces as a consequence of the study of the acoustic tensor, which, in fact, represents the propagation condition of such waves. The surfaces associated with wave front surfaces are slowness surfaces, with slowness defined by the inverse of the wave front speeds, and energy flow surfaces, also known as group velocity surfaces. Here slowness surfaces are studied analytically and numerically to obtain information about wave propagation in arbitrary directions. The degree of wave surface deviation depends upon degrees of anisotropy, and may give valuable information about dynamic deformations. The materials, used in the present analysis, are fibre-reinforced materials with two families of continuous elastic fibres. Since fibres are much stronger than a matrix, anisotropic properties are emphasized. Coordinate-free constitutive relations give the possibility of considering fibres embedded in such materials as extensible or inextensible. Such an approach gives the opportunity to follow behaviour directly considering both the geometry and the stiffness of the fibres. The material considered here is reinforced by two families of mechanically equivalent fibres and, therefore, behaves like orthotropic, having axes of symmetry along bisectors of the fibre directions and along the normal to the plane tangent to the fibres. Such a material has nine independent material constants. When constraints of inextensibility are imposed on the fibres, it leads to the so-called ideal fibre-reinforced material, which may be obtained from materials with extensible fibres in a limiting process. In order to follow the influence of fibre direction on the mechanical properties, constitutive relations are modelled as a function of the fibre directions in each point of the continuous media. An often-used representation of such materials is an epoxy resin–carbon fibre composite whose material constants, for materials reinforced by one family of fibres, are determined using ultrasound methods. Here we consider the dynamical behaviour of the material reinforced by two families of strong mechanically equivalent elastic fibres by comparing the behaviour of extensible fibres to the behaviour of inextensible fibres. Solids made of the above-described materials show some interesting dynamic properties in the propagation of elastic waves. Some recent results of the dynamic behaviour when the material has strong anisotropy shall be presented.

Ray-direction velocities in VTI media

Two-point ray tracing in anisotropic media requires the group and phase velocities to be calculated along ray directions available at each step of a ray bending algorithm. This computation, usually done iteratively or through velocity tables, becomes exceedingly involved for shear-waves that have multivalued group-velocity surfaces, such as in the presence of triplications on the SV wavefronts in vertically transversely isotropic (VTI) media. The difficulties encountered in computing the SV-wave velocities for a given ray direction can be circumvented by solving a polynomial equation whose real-valued roots provide the phase directions of the P- and either one or three SV-waves propagating along a selected ray; those phase directions then allow the group and phase velocities to be computed in a standard fashion. I construct the polynomial and supply computer codes implementing its solution, the codes that can be used in two-point ray-tracing software to improve its performance.

Point source in a phononic grating: stop bands give rise to phonon-focusing caustics

We use locally-excited gigahertz surface phonon wavepackets in microscopic line structures of different pitches to reveal profound anisotropy in the radiation pattern of a point source in a grating. Time-domain data obtained by an ultrafast optical imaging technique and by numerical simulations are Fourier transformed to obtain frequency-filtered real-space acoustic field patterns and k-space phononic band structure. The numerically-obtained k-space images are processed to reveal an intriguing double-horn structure in the lowest-order group-velocity surface, which explains the observed non-propagation sectors bounded by caustics, noted at frequencies above the bottom of the first stop band. We account for these phonon-focusing effects, analogous to collimation effects previously observed in two- and three-dimensional lattices, with a simple analytical model of the band structure based on a plane wave expansion. As the frequency is increased, a transition to dominant waveguiding effects along the lines is also documented.

Temporal variability of the standing internal tide in the Browse Basin, Western Australia

We have investigated the seasonal evolution of the internal tide using a yearlong mooring record in the Browse Basin on the Australian North‐West Shelf. Using a modal harmonic‐decomposition technique, we studied the modulation of the semi‐diurnal kinetic energy, potential energy, energy flux, group velocity and sea surface height anomaly for different internal wave modes. A very coherent and energetic, locally generated internal tide dominated the mooring record during spring‐tide periods. Relationships amongst the wave variables suggest that the interaction of multiple waves formed a standing wave pattern. To assist in explaining the seasonal variability of this standing wave pattern, we used an analytical model of internal wave generation and propagation in a double‐shelf system, representative of the area, to highlight the effects of changes in the stratification and barotropic forcing. A comparison of the low‐mode wave energy indicated that the analytical solution overestimated the observed wave energy by a factor of two and failed to detect an apparent seasonal shift in the conversion rate. We discuss potential mechanisms for this discrepancy.

Anomalous polarization in anisotropic media

Elastic waves can, in principle, be classified according to their propagation velocity, e.g., through the values of the slowness or the group velocity surfaces along a given direction. We show that there are media with the same velocity distribution but drastically different polarization behaviour. Such media are kinematically identical but dynamically different. Therefore, classification according to wave velocity alone is not sufficient, and the identification of the wave type should be based on both velocity and the polarization distribution. For transversely isotropic symmetry there are at most two and for orthorhombic symmetry at most four media that have the same velocity distribution. There is always one medium with a polarization distribution topologically similar to that of isotropy, which is called “normal polarization”. All the other media are said to possess “anomalous polarization”. In orthorhombic media there is a different set of four media closely related to the above mentioned set. The two sets share all velocities in the symmetry planes, but the velocity distribution off the planes is different in the two sets. All members of this set possess anomalous polarization. It is shown that anomalous companions also exist in media of monoclinic symmetry, and we find the corresponding conditions on the elastic constants. The stiffness matrix is that of an orthorhombic “root” medium with an anomalous companion, but with the symmetry broken through the addition of either c14 or c25 or c36 (two of the three stiffnesses must vanish, with one of the three stiffnesses c14, c25 or c36 added). Note that this analysis does not imply that anomalous media could be found in nature, but only that they are not forbidden by the laws of physics.

Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-ir ranges

We discuss propagation effects in realistic, transparent, metallodielectric photonic band gap structures in the context of negative refraction and super-resolution in the visible and near infrared ranges. In the resonance tunneling regime, we find that for transverse-magnetic incident polarization, field localization effects contribute to a waveguiding phenomenon that makes it possible for the light to remain confined within a small fraction of a wavelength, without any transverse boundaries, due to the suppression of diffraction. This effect is related to negative refraction of the Poynting vector inside each metal layer, balanced by normal refraction inside the adjacent dielectric layer: The degree of field localization and material dispersion together determine the total momentum that resides within any given layer, and thus the direction of energy flow. We find that the transport of evanescent wave vectors is mediated by the excitation of quasistationary, low group velocity surface waves responsible for relatively large losses. As representative examples we consider transparent metallodielectric stacks such as $\mathrm{Ag}∕{\mathrm{TiO}}_{2}$ and $\mathrm{Ag}∕\mathrm{GaP}$ and show in detail how to obtain the optimum conditions for high transmittance of both propagating and evanescent modes for super-guiding and super-resolution applications across the visible and near IR ranges. Finally, we study the influence of gain on super-resolution. We find that the introduction of gain can compensate the losses caused by the excitation of surface plasmons, improves the resolving characteristics of the lens, and leads to gain-tunable super-resolution.

Magnetohydrodynamic waves in non-uniform flows I: a variational approach

The interaction of magnetohydrodynamic (MHD) waves in a non- uniform, time-dependent background plasma flow is investigated using Lagrangian field theory methods. The analysis uses Lagrangian maps, in which the position of the fluid element x ∗ is expressed as a vector sum of the position vector x of the background plasma fluid element plus a Lagrangian displacement ξ(x ,t ) due to the waves. Linear, non-Wentzel-Kramer-Brillouin (WKB) wave interaction equations are obtained by expansion of the Lagrangian out to second order in ξ and ∆S, where ∆S is the Lagrangian entropy perturbation. The characteristic manifolds of the waves are determined by consideration of the Cauchy problem for the wave interaction equations. The manifolds correspond to the usual MHD waves modes, namely the Alfvwaves, the fast and slow magnetoacoustic waves and the entropy wave. The relationships between the characteristic manifolds, and the ray equations of geometrical MHD optics are developed using the theory of Cauchy characteristics for first-order partial differential equations. The first-order differential equations describing the singular manifolds are the dispersion equations for the MHD ei- genmodes, where the wave vector k = ∇φ and frequency ω = −φt correspond to the characteristic manifolds φ(x ,t )= constant. The form of the characteristic manifolds for both time-dependent and steady MHD flows are developed. The bi- characteristics for steady MHD waves in a steady background flow are related to the group velocity surface and Mach cone for the waves, and determine when the flow is elliptic, hyperbolic, or of mixed hyperbolic-elliptic type. The wave interaction equations are decomposed into coupled equations for the compressible and incompressible perturbations.

Negative refraction of phonons and acoustic lensing effect of a crystalline slab

We study how good a flat slab of a bulk crystalline solid with a large elastic anisotropy exhibits a lensing action for phonons or sound waves. The slowness and group-velocity surfaces of an ideal elastic solid for a flat phonon lens are analyzed in the geometrical acoustic approximation. These surfaces are compared with the corresponding surfaces of an existing bulk crystal (a zinc crystal) with hexagonal symmetry. To demonstrate the lensing effect we calculate the intensity distribution of phonons emitted from a point source in an isotropic medium (on one side of the lens), propagating through the slab lens and then transmitted into the isotropic medium in the other side. A similar calculation for sound waves with a finite-difference-time-domain method is performed to see the effects neglected in the geometrical acoustic approximation, that is, the effects of finite wavelength, mode conversion, and finite transmission at the interfaces.

Dispersion and characteristic surfaces of waves in hybrid multilayered piezoelectric circular cylinders

The frequency and group velocity dispersion behaviors, and characteristic surfaces of waves in a hybrid multilayered piezoelectric circular cylinder are investigated. The associated frequency dispersion equation is developed using an analytical-numerical method. In this method, the cylinder is modeled using the three-nodal-line layer element; the coupling between the elastic field and the electric field is considered in each element. A system of governing differential equations of each layer element is obtained following the Hamilton Principle. The phase velocity and slowness as well as the group velocity and slowness are established in terms of the Rayleigh quotient. Six characteristic wave surfaces, e.g. the phase velocity, slowness and wave surfaces as well as the group velocity, slowness and wave surfaces, are introduced to visualize the effects of anisotropy and piezoelectricity on wave propagation. A corresponding program code is developed and numerical examples are presented for hybrid multilayered piezoelectric circular cylinders with two ratios of radius to thickness.

Improved First-Order Approximation of Group Velocities in Weakly Anisotropic Media

A first-order approximation of the group velocity is derived for qP and qS waves in weakly anisotropic media. The formula gives an explicit expression of the group velocity in terms of elastic parameters and wave normal and is independent of any reference isotropic media. The approximated group velocity differs from the first order phase velocity in direction and in magnitude, the difference being of the first order in direction and the second order in magnitude. The accuracy of the approximate formula is tested on two examples of TI media. The formula well approximates the qS-waves group velocity surface even in the presence of triplications.

Dispersion of waves and characteristic wave surfaces in functionally graded piezoelectric plates

An inhomogeneous layer element method is presented to analyze the dispersion of waves and characteristic wave surfaces in plates of functionally graded piezoelectric material (FGPM). In this method, the FGPM plate is divided into a number of layered elements. The elemental elastic and electric properties are assumed as linear functions of the thickness to adopt the variety of the material property of FGPM. The Hamilton principle is applied to determine the governing equations. The phase velocity surface, phase slowness surface, phase wave surface, group velocity surface, group slowness surface, and group wave surface for FGPM plate are formulated using Rayleigh quotient and the orthogonality condition of the eigenvectors. These six surfaces are then used to illustrate the characteristics of waves in FGPM plates. Numerical examples are presented using the present formulations to analyze dispersions and characteristics of waves in FGPM plates.

Anisotropy correction for deviated‐well sonic logs: Application to seismic well tie

Borehole sonic logs acquired in deviated wells penetrating the HRZ and Colville shales in the Niakuk field in Alaska's North Slope are seen to be significantly faster than vertical well logs. These differences are attributed to shale anisotropy. An iterative inversion scheme was created to invert for shale anisotropy parameters using multiple wells penetrating shale sections at different angles. The inversion involves fitting the sonic log data at a range of borehole angles to the compressional wave group velocity surface. The result is an estimate of the anisotropy parameters (ε and δ) and the vertical P‐wave velocity. The results show that the shales are strongly anisotropic, with compressional‐wave anisotropy (Thomsen's parameter ε) on the order of 40% and the anisotropy parameter δ (relates vertical velocity to short‐offset NMO velocity) around 10%. This large anisotropy can affect seismic imaging, AVO, and time–depth calculations.A procedure was created to estimate the anisotropy‐corrected vertical sonic logs from sonic data recorded in a deviated well. The inputs are well deviation, P‐wave sonic log, volume of shale or gamma ray data, and anisotropy parameters for rock with 100% shale volume. With these inputs the compressional‐wave group velocity surface is computed and the equivalent vertical P‐wave sonic log is output. The equivalent vertical sonic log can then be used for standard seismic applications using isotropic velocity assumptions. The application was applied to a well deviated at approximately 67°. Shale anisotropy parameters were taken from the sonic log inversion, and an anisotropy‐corrected sonic log was produced. Seismic well ties were attempted using both the recorded logs and the anisotropy‐corrected logs, with the result that the well tie using the measured logs was poor while a tie using the anisotropy‐corrected logs was good.

Focusing of light in a three-dimensional cubic photonic crystal

The dielectric properties of photonic crystals are, in general, anisotropic spatially. This means that the constant-frequency surfaces of photons in wave-vector space are deformed from spherical shapes, leading to the photon focusing effects which are significantly enhanced at frequencies close to the photonic band gaps (PBG's). We find that the associated group-velocity surfaces of photons in real space are folded near PBG's and this gives rise to the appearance of photon caustics. We explicitly show these effects by illustrating theoretical images of photons transmitted through a photonic crystal. The calculation is made with the plane-wave-expansion method and numerical examples are given for a three-dimensional cubic lattice consisting of air and silicon.

Fermat's Variational Principle for Anisotropic Inhomogeneous Media

Fermat's variational principle states that the signal propagates from point S to R along a curve which renders Fermat's functional \(\mathcal{I}\)(l) stationary. Fermat's functional \(\mathcal{I}\)(l) depends on curves l which connect points S and R, and represents the travel times from S to R along l. In seismology, it is mostly expressed by the integral \(\mathcal{I}\)(l) = \(\smallint _S^R \mathcal{L}\)(xk,xk')du, taken along curve l, where ℒ(xk,xk') is the relevant Lagrangian, xk are coordinates, u is a parameter used to specify the position of points along l, and xk' = dxk÷du. If Lagrangian ℒ(xk,xk') is a homogeneous function of the first degree in xk', Fermat's principle is valid for arbitrary monotonic parameter u. We than speak of the first-degree Lagrangian ℒ(1)(xk,xk'). It is shown that the conventional Legendre transform cannot be applied to the first-degree Lagrangian ℒ(1)(xk,xk') to derive the relevant Hamiltonian ℋ(1)(xk,pk), and Hamiltonian ray equations. The reason is that the Hessian determinant of the transform vanishes identically for first-degree Lagrangians ℒ(1)(xk,xk'). The Lagrangians must be modified so that the Hessian determinant is different from zero. A modification to overcome this difficulty is proposed in this article, and is based on second-degree Lagrangians ℒ(2). Parameter u along the curves is taken to correspond to travel time τ, and the second-degree Lagrangian ℒ(2)(xk,\(\dot x\)k) is then introduced by the relation ℒ(2)(xk,\(\dot x\)k) = [ℒ(1)(xk,\(\dot x\)k)]2, with \(\dot x\)k = dxk÷dτ. The second-degree Lagrangian ℒ(2)(xk,\(\dot x\)k) yields the same Euler/Lagrange equations for rays as the first-degree Lagrangian ℒ(1)(xk,\(\dot x\)k). The relevant Hessian determinant, however, does not vanish identically. Consequently, the Legendre transform can then be used to compute Hamiltonian ℋ(2)(xk,pk) from Lagrangian ℒ(2)(xk,\(\dot x\)k), and vice versa, and the Hamiltonian canonical equations can be derived from the Euler-Lagrange equations. Both ℒ(2)(xk,\(\dot x\)k) and ℋ(2)(xk,pk) can be expressed in terms of the wave propagation metric tensor gij(xk,\(\dot x\)k), which depends not only on position xk, but also on the direction of vector \(\dot x\)k. It is defined in a Finsler space, in which the distance is measured by the travel time. It is shown that the standard form of the Hamiltonian, derived from the elastodynamic equation and representing the eikonal equation, which has been broadly used in the seismic ray method, corresponds to the second-degree Lagrangian ℒ(2)(xk,\(\dot x\)k), not to the first-degree Lagrangian ℒ(1)(xk,\(\dot x\)k). It is also shown that relations ℒ(2)(xk,\(\dot x\)k) = \( - \frac{1}{2}\); and ℋ(2)(xk,pk) = \( - \frac{1}{2}\) are valid at any point of the ray and that they represent the group velocity surface and the slowness surface, respectively. All procedures and derived equations are valid for general anisotropic inhomogeneous media, and for general curvilinear coordinates xi. To make certain procedures and equations more transparent and objective, the simpler cases of isotropic and ellipsoidally anisotropic media are briefly discussed as special cases.

Laboratory results for the features of body‐wave propagation in a transversely isotropic media

Much of the earth’s crust appears to have some degree of elastic anisotropy (Crampin, 1981; Crampin and Lovell, 1991; Helbig, 1993). The phenomena of elastic wave propagation in anisotropic media are more complex than those in isotropic media. Shear‐wave propagation in an orthorhombic physical model is most complex when the direction of the wave is close to the neighborhood of the cusp on the group velocity surfaces (Brown et al., 1991). The first identification of singularities in wave propagation through sedimentary basins occurred in the examination of shear‐wave splitting in multioffset vertical seismic profiles (VSPs) at a borehole site in the Paris Basin (Bush and Crampin, 1991), where large variations in shear‐wave polarizations in propagation directions close to point singularities were observed. Computation of synthetic seismograms for layer sequences showed that the shear‐wave polarizations and amplitudes were irregular near point singularities (Crampin, 1991).

Dispersion and characteristic surfaces of waves in laminated composite circular cylindrical shells

The dispersion behaviors and characteristic surfaces of waves in a laminated composite circular cylindrical shell are investigated using a semianalytical method based on the theory of three-dimensional elasticity. The radial displacement of the shell is modeled by finite elements, while the axial and circumferential displacements are expanded as the complex exponentials. The associated characteristic equation is developed by means of the Hamilton's principle. The eigenvalues are established in terms of the Rayleigh quotient. Six characteristic wave surfaces, viz., the phase velocity, phase slowness, and phase wave surfaces, as well as the group velocity, group slowness, and group wave surfaces, are introduced to visualize the effects of anisotropy on wave propagation. Numerical examples demonstrate that the ratio of the inner radius to the thickness of the shell has a stronger influence on the frequency spectra in the circumferential wave than on that in the axial wave; that negative group velocity appears at a range of smaller wave numbers and the range varies as the wave normal and the ratio of the inner radius to the thickness of the shell; and that the characteristic wave surfaces vary with the propagation modes of waves, the ratio of the inner radius to the thickness of the shell, and the lay-ups of the laminated shells.

Angular spectrum approach for the computation of group and phase velocity surfaces of acoustic waves in anisotropic materials

The decomposition of an acoustic wave into its angular spectrum representation creates an effective base for the calculation of wave propagation effects in anisotropic media. In this method, the distribution of acoustic fields is calculated in arbitrary planes from the superposition of the planar components with proper phase shifts. These phase shifts depend on the ratio of the distance between the planes to the normal component of the phase slowness vector. In anisotropic media, the phase shifts depend additionally on the changes of the slowness with respect to the direction of the propagation vector and the polarization. Those relations are obtained from the Christoffel equation. The method employing the fast Fourier transformation algorithm is especially suited for volume imaging in anisotropic media, based on holographic detection in transmission of acoustic waves generated by a point source. This technique is compared with measurements on crystals performed by phase-sensitive scanning acoustic microscopy.

Determination of the group and phase velocity surfaces of acoustic waves in anisotropic materials by ultrasonic acoustic microscopy with phase contrast

A new method to determine the group and phase velocity surfaces using only one image in phase contrast detected by phase sensitive scanning acoustic microscopy (PSAM) has been developed. Similar to earlier techniques, a focusing ultrasonic transducer is adjusted to a spot on a surface of a sample. A second similar transducer is scanned with the focus in the plane of the opposing planar surface. In the application presented here the group and phase velocity surfaces are derived from the observed phases of the transmitted ultrasonic waves. Numerical methods have been developed to achieve a continuous representation of the phases from the measured images. Subsequently the data is corrected for geometrical effects caused by the planar scanning sheet leading to a three-dimensional representation of the velocity surfaces. By additional procedures based on simple geometrical conditions, the phase velocities are derived in a straightforward way from the group velocities.

Factoring anisotropy into iterative geometric reconstruction algorithms for seismic tomography

Tomography algorithms based on isotropic velocities need upgrading to perform well in anisotropic materials. In this paper, anisotropy is introduced in a SIRT-type algorithm in two ways. First the group velocity surfaces are assumed to be elliptical. Then the first order approximation of the velocity for transversely isotropic material is factored into the algorithm. It is shown that the latter modification which is less restrictive than elliptical anisotropy, can be conveniently inserted into existing algorithms while preserving their simplicity and calculation time. In addition to the mathematical formalism, applications with field data are discussed.