Wave velocities in a pre-stressed anisotropic elastic medium

Studying the anisotropy of seismic wave velocity can provide a theoretical basis for understanding the characteristics of seismic wave propagation in anisotropic media, such as forward modeling of seismic wavefield and simultaneous inversion of anisotropic parameters. It is an important work to calculate the group velocities of three kinds of seismic waves (qP, qS1, and qS2) and the analytical expressions of the partial derivatives of the group velocities with respect to 21 elastic parameters in general anisotropic media. Therefore, a new relationship between the phase and group velocity in the general anisotropic medium is introduced first, and then the analytical expressions of the group velocity of three kinds of waves are deduced. Then, the analytical formulas of partial derivatives of the group (or phase) velocity with respect to 21 elastic parameters are derived. Finally, the distribution of partial derivatives of group slowness with respect to 21 elastic parameters with varied ray angles is analyzed and discussed. The final analytical expressions of group velocity and its partial derivatives are relatively concise and can deal with the singular points and multivalued problems of qS waves. By drawing comparison to the numerical solution of the eigenvector method, the accuracy of the analytical formula of group velocity and its partial derivatives with respect to elastic parameters are verified. The numerical simulation results indicate that the algorithm can accurately obtain the partial derivatives of group slowness with respect to 21 elastic parameters and would provide a theoretical basis for seismic ray tracing and traveltime inversion in general anisotropic media.

The effect of initial stress on the propagation behavior of SH waves in piezoelectric coupled plates

This paper presents the Lebedev scheme on staggered grids for the numerical simulation of wave propagation in anisotropic elastic media. Primary attention is given to the approximation of the elastic wave equation by the Lebedev scheme. Based on the differential approach, it is shown that the Lebedev scheme approximates a system of equations, which differs from the original equation. It is proved that the approximated system has a set of 24 characteristics, six of them coincide with those of the elastic wave equation and the rest ones are “artifacts.” Requiring the artificial solutions to be equal to zero and the true ones to coincide with those of the elastic wave equation, one comes to the classical definition of the approximation of the initial system on a sufficiently smooth solution. The results obtained and the knowledge of the complete set of characteristics are important for constructing reflectionless boundary conditions during approximation of point sources, etc.

The paper deals with finite–difference (f‐d) approach to simulation of elastic waves' propagation in anisotropic elastic media with general symmetry. Any implementation of this approach claims resolution of two key problems: construction of an effective f‐d scheme itself; we propose to use the Lebedev's scheme (LS) being a natural generalization of Virieux staggered grid scheme (VS) widely used for isotropy; we prove that LS possesses better dispersion properties in comparison with well known Rotated Staggered Grids Scheme (RSGS). stable domain distension. The Perfectly Matched Layers(PML) useful for isotropic problems can be unstable in the case of anisotropy. Lebedev scheme allows one to use Optimal Grids (OG) which gives a possibility to implement efficient and low cost domain distension. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Traveltime computation is an effective way to simulate seismic wave propagation in isotropic and anisotropic media. It often requires the phase and group velocities along a ray direction. The phase and group velocities are not functions of the ray direction, but functions of the slowness direction, which motivates the need for efficient numerical approaches for computing the slowness direction. A generalized method was proposed to achieve this goal in 3D tilted transverse isotropic media, which involves solving a nonlinear equation of two unknowns, the inclination and azimuthal angles of the slowness direction, and is computationally demanding. To overcome the difficulty, the equation was recast by projecting it onto the local coordinates that align with the axis of symmetry of the medium. Under the local coordinate system, the nonlinear equation reduces to an equation of one unknown, the inclination angle, with the azimuthal angle obtained from the given ray direction. Hence, the nonlinear equation can be solved efficiently and its solutions can be used to calculate the phase and group velocities. As an application, the proposed method was used to compute traveltimes in 3D vertical transverse isotropic (VTI) media. The calculated group velocities are incorporated into the fast sweeping method to solve the original anisotropic Eikonal equation directly on uniform meshes. The feasibility of the proposed group velocity calculation method is evaluated in three 3D homogeneous anisotropic models, and the application on traveltime computation is tested on a 3D homogeneous VTI model and the British Petroleum anisotropic VTI model.

Dynamic stiffness of three-dimensional anisotropic multi-layered media based on the continued-fraction method

Propagation of torsional surface wave in anisotropic poroelastic medium under initial stress

ABSTRACTIn this study, the characteristics of Love waves in viscoelastic vertical transversely isotropic layered media are investigated by finite-difference numerical modeling. The accuracy of the modeling scheme is tested against the theoretical seismograms of isotropic-elastic and isotropic-viscoelastic media. The correctness of the modeling results is verified by the theoretical phase-velocity dispersion curves of Love waves in isotropic or anisotropic elastic or viscoelastic media. In two-layer half-space models, the effects of velocity anisotropy, viscoelasticity, and attenuation anisotropy of media on Love waves are studied in detail by comparing the modeling results obtained for anisotropic-elastic, isotropic-viscoelastic, and anisotropic-viscoelastic media with those obtained for isotropic-elastic media. Then, Love waves in three typical four-layer half-space models are simulated to further analyze the characteristics of Love waves in anisotropic-viscoelastic layered media. The results show that Love waves propagating in anisotropic-viscoelastic media are affected by both the anisotropy and viscoelasticity of media. The velocity anisotropy of media causes substantial changes in the values and distribution range of phase velocities of Love waves. The viscoelasticity of media leads to the amplitude attenuation and phase velocity dispersion of Love waves, and these effects increase with decreasing quality factors. The attenuation anisotropy of media indicates that the viscoelasticity degree of media is direction dependent. Comparisons of phase velocity ratios suggest that the change degree of Love-wave phase velocities due to viscoelasticity is much less than that caused by velocity anisotropy.

Seismic waves propagating in anisotropic earth media suffer from waveform distortion, which leads to degraded resolution in seismic data imaging if this adverse effect is not corrected. Additionally, the wavefields simulated by the traditional coupled pseudo‐acoustic wave equations contain undesired shear wave artefacts and are unstable when Thomsen's anisotropy parameter ε is less than δ. To address these issues, many pure qP‐wave equations are proposed to describe wave propagation in anisotropic media. However, these equations are either low precision or high accuracy but computationally expensive. Considering the trade‐off between the computation efficiency and simulation accuracy, we derive a pure qP‐wave equation from the exact dispersion formula in tilted transverse isotropic media. The newly derived equation has only two higher order partial derivatives, which can better balance the accuracy and efficiency than the previous pure qP‐wave equations. The least‐squares reverse time migration method can balance amplitudes, suppress migration artefacts and improve imaging resolution. Then, based on the newly derived pure qP‐wave equation, we derive the Born modelling operator and adjoint migration operator and develop an anisotropic least‐squares reverse time migration method. To achieve the accelerated convergence, we introduce a generalized minimum residual algorithm to implement the anisotropic least‐squares reverse time migration method. Numerical simulation results show that the proposed pure qP‐wave equation outperforms the previous equations in balancing the trade‐off between accuracy and efficiency. In addition, synthetic examples demonstrate the effectiveness and robustness of our proposed anisotropic least‐squares reverse time migration method in correcting the deviations due to anisotropic effects.

PurposeThe purpose of this paper is to study the effect of rotation and initial stress on magneto-thermoelastic material with voids heated by a laser pulse heating.Design/methodology/approachThe analytical method used was the normal mode analysis technique.FindingsNumerical results for the physical quantities were presented graphically and analyzed. The graphical results indicate that the effect of rotation, initial stress and magnetic fields are observable physical effects on the thermoelastic material with voids heated by a laser pulse. Comparisons are made with the results in the absence and the presence of the physical operators, also at various times.Originality/valueIn the present work, the authors shall investigate the effect of the rotation, initial stress, magnetic field and laser pulse on thermoelastic material with voids subjected to a laser pulse heating acting as a thermal shock. A comparison is also made between the two types (types II and III) of Green-Naghdi theory in the absence and the presence of the physical operators. Such problems are very important in many dynamical systems.

We study the features of propagation of a longitudinal wave in an acoustic (mechanical) metamaterial, modeled as a one-dimensional chain, containing equal masses, connected by elastic elements (springs), and having the same rigidity. Each mass contains within itself a series connection of another mass and viscous element (damper). The mass-to-mass model is free from the drawbacks of a number of other mechanical models of metamaterials: i.e. it eliminates the need to have the property of a deformable body to possess a negative mass, density, and (or) a negative elastic modulus. It is shown that the model under consideration makes it possible to describe the dispersion and frequency-dependent attenuation of a longitudinal wave, the character of which essentially depends on the ratio of the external and internal mass of the metamaterial. The behavior of the phase and group velocities of the wave is studied, as well as the evolution of its profile, both in the low-frequency and high-frequency ranges. The mass ratios were found at which the phase velocity exceeds the group velocity (normal dispersion) in magnitude and those at which the group velocity exceeds the phase velocity (anomalous dispersion) in a wide frequency range. Having the same asymptotic values when the frequency tends to infinity, the phase and group velocities have significant differences in behavior, namely, that the phase velocity is a monotonic function of frequency, and the group velocity has a maximum. In addition, in the region of normal dispersion, the group velocity may be negative, i.e. the so-called “reverse wave” effect is true, when, despite the fact that the phase velocity is directed in the positive direction of the spatial axis, the energy in such a wave is transferred in the negative direction.

I derive analytical expressions giving traveltime‐offset curves for multilayered weakly azimuthally isotropic and anisotropic media on terms of the elastic properties of each layer. This method is based upon an approximate skewed hyperbolic moveout formula involving three measured bulk velocities for each reflector. The primary benefits of this technique are: (1) it allows for fast traveltime computation; (2) it makes possible an extremely rapid estimation of the interval elastic parameters; and (3) it provides physical insight into the wave propagation in anisotropic media. I establish a traveltime inversion algorithm which allows the estimation of the five elastic constants together with the orientation of the axis of symmetry for each layer. In the isotropic limit, this algorithm reduces to the conventional one used for determining the interval velocities from stacking velocity measurements in isotropic media. The inversion technique is applied to surface seismic measurements and VSP field surveys. In both cases, the method provides very good estimates of these six parameters, even in highly anisotropic materials. The results also show that the orientation of the horizontal axis of symmetry, for a given azimuthally anisotropic layer at depth, can be obtained using only quasi‐P‐wave information. This technique can be easily incorporated into conventional velocity analysis algorithms. Interval anisotropic properties can then be estimated with little additional effort.

Plane electromagnetic as well as plane elastodynamic waves in anisotropic media exhibit a different direction of their phase and energy propagation, resulting in slowness and group velocity surfaces. Of course, the availability of plane wave solutions gives rise to a spectral plane wave decomposition of point source excitations, i.e., Green's functions. Unfortunately, the coordinate‐free closed‐form solution of dyadic (electric) Green's functions in the Rω space is only known for electromagnetic (generalized) uniaxial media. Utilizing the relation between phase and group velocities of plane waves in uniaxial media we have been able to show that the phase and amplitude of the Green's function is related to the group velocity; i.e., time domain wave fronts reproduce group velocity surfaces. This has also been verified through numerical results obtained by the three‐dimensional (3‐D) electromagnetic finite integration technique code. In elastodynamics, where similar analytical results for anisotropic media are not available, we confirm this behavior with our numerical 3‐D elastodynamic finite integration technique code. For electromagnetic uniaxial media, we present an analytic method to derive the dyadic far‐field Green's function in Rω space from Kω space directly by utilizing the duality principle between wave vectors and ray vectors without performing the 3‐D inverse Fourier transform from Kω space to Rω space analytically.

Propagation of acoustic waves at a high frequency in anisotropic media is considered. In this case, the WKB approximation results in eikonal equations whose Hamiltonians are neither convex nor concave in the impulse variable as it is the case in differential games theory. In this paper the methods of differential games are adopted for the analysis of wave propagation. If the Hamiltonian of a differential game approximates the Hamiltonian of the eikonal equation, then the solution to the game approximates the phase function satisfying the eikonal equation. The method of singular characteristics is used for the analysis of singularities in associated differential games. Numerical results are presented for wave velocity surfaces typical for anisotropic quartz crystals.

The method of perfectly matched layers (PML) is based on the elegant idea of introducing attenuation while keeping impedance unaltered, thus resulting in effective absorption of waves without spurious reflections. Due to its simplicity and effectiveness, PML has been widely used over the past 25 years to model wave propagation in unbounded domains. However, an acknowledged challenge is that PML no longer preserves impedance once the domain and the PML region are discretized, leading to spurious reflection at the interface. This issue has been fixed with the help of so-called impedance preserving discretization, leading to the method of perfectly matched discrete layers (PMDL), developed more than a decade ago. The idea is based on the observation that midpoint integration completely eliminates the discretization error in impedance that arises from linear finite element discretization. Beyond eliminating reflections at the interface, impedance preserving discretization is shown to be crucial to achieving stability of absorbing boundary conditions in situations with differing signs of phase and group velocities, e.g., anisotropic media. This talk will be an exposition of the development of the PMDL by the author and his collaborators, staring with fundamental ideas impedance preserving discretization followed by more advanced ABCs for anisotropic and periodic media.