Wavefronts in dissipative anisotropic media

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.

We study behavior of attenuation (inhomogeneity) angles γ, i.e., angles between real and imaginary parts of the slowness vectors of inhomogeneous plane waves propagating in isotropic or anisotropic, perfectly elastic or viscoelastic, unbounded media. The angle γ never exceeds the boundary attenuation angle γ*. In isotropic viscoelastic media γ* = 90°; in anisotropic viscoelastic media γ* may be greater than, equal to, or less than 90°. Plane waves with γ > γ* do not exist. Because γ* in anisotropic viscoelastic media is usually not known a priori, the commonly used specification of an inhomogeneous plane wave by the attenuation angle γ may lead to serious problems. If γ is chosen close to γ* or even larger, indeterminate, unstable or even nonphysical results are obtained. We study properties of γ* and show that the approach based on the mixed specification of the slowness vector fully avoids the problems mentioned above. The approach allows exact determination of γ* and removes instabilities known from the use of the specification of the slowness vector by γ. For γ = γ*, the approach yields zero phase velocity, i.e., the corresponding wave is a nonpropagating wave mode. The use of the mixed specification leads to the explanation of the deviation of γ* from 90° as a consequence of different orientations of energy-flux and propagation vectors in anisotropic media. The approach is universal; it may be used for isotropic or anisotropic, perfectly elastic or viscoelastic media, and for homogeneous and inhomogeneous waves, including strongly inhomogeneous waves, like evanescent waves.

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].

Modified Christoffel equations are derived for three-dimensional wave propagation in a general anisotropic medium under initial stress. The three roots of a cubic equation define the phase velocities of three quasi-waves in the medium. Analytical expressions are used to calculate the directional derivatives of phase velocities. These derivatives are, further, used to calculate the group velocities and ray directions of the three quasi-waves in a pre-stressed anisotropic medium. Effect of initial stress on wave propagation is observed through the deviations in phase velocity, group velocity and ray direction for each of the quasi-waves. The variations of these deviations with the phase direction are plotted for a numerical model of general anisotropic medium with triclinic/ monoclinic/orthorhombic symmetry

The singularity points are very important for elastic waves propagation in low-symmetry anisotropic media (Stovas et al., 2021a). Being converted into the group velocity domain, they result in internal refraction cone with anomalous amplitudes and very complicated polarization fields. In elastic orthorhombic (ORT) media, there is always one singularity point in the essential symmetry plane, (0,1,2) singularity points in remaining non-essential symmetry planes and (0,1) points in-between the symmetry planes (Stovas et al., 2021b).I analyze the conditions for existence of a singularity point in-between the symmetry planes.In order to do that I fix the diagonal elements of the stiffness coefficient matrix, cjj , j=1,6, and introduce new variables d12 = c12 +c66, d13 = c13 +c55 and d23 = c23 +c44. I also assume that the symmetry plane 2-3 is the essential one by introducing the inequality c55 < c44 < c66 . If c66 < c44 < c55, the 2-3 plane is still essential one but the properties of non-essential planes will interchange. In case of other inequalities for “S wave” stiffness coefficients, the corresponding properties of singularity points can be obtained by a cyclic rotation of stiffness coefficients and symmetry planes (Stovas et al., 2023).For selected essential symmetry plane (2-3), I propose to fix two variables d12 and d13, and set the variable d23 as a free variable. By changing d23 only, I can define the trajectory of a singularity point in-between the symmetry planes. This trajectory is given by a continuous line connecting the symmetry planes. Then I define the traces of this trajectory on symmetry planes (maximum two points for each plane) by a specific value of the variable d23. These 6 values can be used for intervals of d23 where the singularity point in-between the symmetry planes exists. Analysis shows that there are 7 zones in (d12 , d13) plane with different intervals of d23, which guarantee the existence of singularity point in-between the symmetry planes. There are 3 intervals of d23 in one zone, two intervals in two zones and one interval in three zones. There is no singularity point in-between the symmetry planes for any d23 in remaining zone. These zones are separated by three straight lines that defined by d12 = d12(critical) , d13 = d13(critical)  and d13 = α d12, where α guarantees that trajectory of singularity point meets the essential symmetry plane. ReferencesStovas, A., Roganov, Yu., and V. Roganov, 2021a, Geometrical characteristics of P and S wave phase and group velocity surfaces in anisotropic media, Geophysical Prospecting, 68(1), 53-69.Stovas, A., Roganov, Yu., and V. Roganov, 2021b, Wave characteristics in elliptical orthorhombic medium, Geophysics, 86(3), C89-C99.Stovas, A., Roganov, Yu., and V. Roganov, 2023, On singularity points in elastic orthorhombic media, Geophysics, 88(1), C11-C32.

The attenuation vector plays an important role in investigating waves propagating in dissipative, isotropic or anisotropic media. It is defined as the imaginary part of the complex-valued traveltime gradient. The real part of the complex-valued traveltime gradient is referred to as the propagation vector. In this paper, simple expressions for the attenuation vector are derived for seismic body waves propagating in heterogeneous, weakly dissipative, anisotropic media. The ray-theory perturbation method is used, in which the dissipative medium is considered to be a small perturbation of a perfectly elastic medium. The general perturbation procedure proposed by Klimeš (2002) is very suitable for this purpose, as it does not require ray tracing in dissipative media; the computation of rays in the reference perfectly elastic medium, followed by quadratures along them, is quite sufficient. It is shown that the waves propagating in a heterogeneous, weakly dissipative, anisotropic or isotropic medium are, in general, inhomogeneous. This means that their attenuation vector is not parallel to the propagation vector. This holds even for waves generated by a point-source in a homogeneous anisotropic weakly dissipative medium. An exception is a wave generated by a point-source in a homogeneous isotropic dissipative medium, where the generated wave is homogeneous. Thus, the commonly used concept of homogeneous waves can be applied neither to heterogeneous nor to anisotropic dissipative media. The situation is different for plane waves propagating in homogeneous dissipative isotropic or anisotropic media. In this case, the homogeneity or inhomogeneity of the plane wave may be chosen freely. Besides the attenuation vector, we also study the complex-valued ray-velocity vector. In heterogeneous media, the complex-valued ray-velocity vector is generally inhomogeneous, that is, its real and imaginary parts are not parallel. It is also shown that twice the scalar product of the attenuation vector with the ray-velocity vector in the reference medium yields 1/Q, where Q is the position- and direction-dependent quality factor. Quality factor Q does not depend on the inhomogeneity of the wave under consideration and offers a convenient measure of intrinsic material dissipation.

SUMMARY We study properties of the energy-flux vector and other related energy quantities of homogeneous and inhomogeneous time-harmonic P and S plane waves, propagating in unbounded viscoelastic anisotropic media, both analytically and numerically. We propose an algorithm for the computation of the energy-flux vector, which can be used for media of unrestricted anisotropy and viscoelasticity, and for arbitrary homogeneous or inhomogeneous plane waves. Basic part of the algorithm is determination of the slowness vector of a homogeneous or inhomogeneous wave, which satisfies certain constraints following from the equation of motion. Approaches for determination of a slowness vector commonly used in viscoelastic isotropic media are usually difficult to use in viscoelastic anisotropic media. Sometimes they may even lead to non-physical solutions. To avoid these problems, we use the so-called mixed specification of the slowness vector, which requires, in a general case, solution of a complex-valued algebraic equation of the sixth degree. For simpler cases, as for SH waves propagating in symmetry planes, the algorithm yields simple analytic solutions. Once the slowness vector is known, determination of energy flux and of other energy quantities is easy. We present numerical examples illustrating the behaviour of the energy-flux vector and other energy quantities, for homogeneous and inhomogeneous plane P, SV and SH waves. 1 I NTR O DUCTION In this paper, we study the properties of the energy flux of homogeneous and inhomogeneous time-harmonic plane waves propagating in an unbounded viscoelastic anisotropic medium. The complexvalued energy-flux vector F (also called the Poynting or Umov‐ Poynting vector) is used as the quantity which fully characterizes the energy flux and the dissipation of energy of the wave. The real part of the vector F represents the time-averaged energy flux S. Both real, Re F = S, and imaginary, Im F, parts are closely related to the density of the time-averaged dissipated energy W d : W d equals twice the scalar product of −Im F and propagation vector P (real part of the slowness vector, perpendicular to the wave front) or twice the scalar product of Re F and attenuation vector A (imaginary part of the slowness vector, perpendicular to the plane of constant amplitude). The aim of this paper is to present a concise derivation of the expression for the complex-valued energy-flux vector F of homogeneous and inhomogeneous plane waves propagating in unbounded media of unrestricted anisotropy and viscoelasticity, to propose an algorithm for its calculation, and to perform analytical and numerical studies necessary for correct understanding of energy propagation and dissipation of the relevant plane waves. Considerable attention in seismology has been devoted to the energy flux of plane waves propagating in elastic anisotropic and in viscoelastic isotropic media. For elastic anisotropic media, see, for

It is proved that if an anisotropic medium has an open set of singular directions, then this medium has two slowness surfaces that completely coincide. The coinciding slowness surfaces form one double singular slowness surface. The corresponding anisotropic medium is an elliptical orthorhombic (ORT) medium with equal stiffness coefficients c44=c55=c66 rotated to an arbitrary coordinate system. Based on the representation of the Christoffel matrix as a uniaxial tensor and considering that the elements of the Christoffel matrix are quadratic forms in the components of the slowness vector, a system of homogeneous polynomial equations was derived. Then, the identical equalities between homogeneous polynomials are replaced by the equalities between their coefficients. As a result, a new system of equations is obtained, the solution of which is the values of the reduced (density normalized) stiffness coefficients in a medium with a singular surface. Conditions for the positive definite of the obtained stiffness matrix are studied. For the defined medium, the Christoffel equations and equations of group velocity surfaces are derived. The orthogonal rotation matrix that transforms the medium with a singular surface into an elliptic ORT medium in the canonical coordinate system is determined. In the canonical coordinate system, the slowness surfaces S1 and S2 waves coincide and are given by a sphere with a radius . The slowness surface of qP waves in the canonical coordinate system is an ellipsoid with semi-axes , , . The polarization vectors of S1 and S2 waves can be arbitrarily selected in the plane orthogonal to the polarization vector of the qP wave. However, the qP wave polarization vector can be significantly different from the wave vector. This feature should be taken into account in the joint processing and modelling of S and qP waves. The results are illustrated in one example of an elliptical ORT medium.

Foreword. I: Theoretical problems concerning propagation of surface wave in elastic anisotropic media. On the role of anisotropy in crystalloacoustics V.I. Alshits. Surface waves of non-Rayleigh type S.V. Kuznetsov. Nonlinearity in elastic surface waves acts nonlocally D.F. Parker. Explicit secular equations for surface waves in an anisotropic elastic half space from Rayleigh to today T.C.T. Ting. II: Bending and edge waves in plates and shells. 'Nongeometrical phenomena' in propagation of elastic surface waves V.M. Babich, A.P. Kiselev. Complex rays and internal diffraction at the cusp edge M. Deschamps, O. Poncelet. Edge waves in the fluid beneath an elastic sheet with linear nonhomogeneity R.V. Goldstein, A.V. Marchenko. On continuum modelling of wave propagation in layered medium: bending waves K.B. Ustinov. Edge-localised bending waves in anisotropic media: energy and dispersion III: Experimental, numerical and semi-analytical methods for analysis of surface waves. Surface electromagnetic perturbations induced by unsteady-state subsurface flow P.M. Adler, V.M. Entov. Resonant waves in a structured elastic halfspace M.V. Ayzenberg-Stepanenko. Mumerical analysis of Rayleigh waves in anisotropic media A.V. Kaptsov. Guided waves in anisotropic media: applications R.A. Kline. A general purpose computer model for calculating elastic waveguide properties, with application to Non-Destructive Testing M.J.S. Lowe, P. Cawley, B.N. Pavlakovic. The influence of the initial stresses on the dynamic instability of an anisotropic cone Y.R. Rossikhin, M.V. Shitikova. Embedding theorem and mutual relation for the interface and shear wavespeeds I.V. Simonov. IV: Applications of surface waves to analysis of fracture and damage. The non-uniqueness of constant velocity crack propagation K.B. Broberg. Embedding formulae for planar cracks R.V. Craster, A.V. Shanin. Wave propagation and crack detection in layered structures E.V. Glushkov, N.V. Glushkova, A.V. Ekhlakov. Author Index.

The development of reliable methods for upscaling fine-scale models of elastic media has long been an important topic for rock physics and applied seismology. Several effective medium theories have been developed to provide elastic parameters for materials such as finely layered media or randomly oriented or aligned fractures. In such cases, the analytic solutions for upscaled properties can be used for accurate prediction of wave propagation. However, such theories cannot be applied directly to homogenize elastic media with more complex, arbitrary spatial heterogeneity. Therefore, we have proposed a numerical homogenization algorithm based on multiscale finite-element methods for simulating elastic wave propagation in heterogeneous, anisotropic elastic media. Specifically, our method used multiscale basis functions obtained from a local linear elasticity problem with appropriately defined boundary conditions. Homogenized, effective medium parameters were then computed using these basis functions, and the approach applied a numerical discretization that was similar to the rotated staggered-grid finite-difference scheme. Comparisons of the results from our method and from conventional, analytical approaches for finely layered media showed that the homogenization reliably estimated elastic parameters for this simple geometry. Additional tests examined anisotropic models with arbitrary spatial heterogeneity in which the average size of the heterogeneities ranged from several centimeters to several meters, and the ratio between the dominant wavelength and the average size of the arbitrary heterogeneities ranged from 10 to 100. Comparisons to finite-difference simulations proved that the numerical homogenization was equally accurate for these complex cases.

A comprehensive approach, based on the general nonlinear ray perturbation theory (Druzhinin, 1991), is proposed for both a fast and accurate uniform asymptotic solution of forward and inverse kinematic problems in anisotropic media. It has been developed to modify the standard ray linearization procedures when they become inconsistent, by providing a predictable truncation error of ray perturbation series. The theoretical background consists in a set of recurrent expressions for the perturbations of all orders for calculating approximately the body wave phase and group velocities, polarization, travel times, ray trajectories, paraxial rays and also the slowness vectors or reflected/transmitted waves in terms of elastic tensor perturbations. We assume that any elastic medium can be used as an unperturbed medium. A total 2-D numerical testing of these expressions has been established within the transverse isotropy to verify the accuracy and convergence of perturbation series when the elastic constants are perturbed. Seismological applications to determine crack-induced anisotropy parameters on VSP travel times for the different wave types in homogeneous and horizontally layered, transversally isotropic and orthorhombic structures are also presented. A number of numerical tests shows that this method is in general stable with respect to the choice of the reference model and the errors in the input data. A proof of uniqueness is provided by an interactive analysis of the sensitivity functions, which are also used for choosing optimum source/receiver locations. Finally, software has been developed for a desktop computer and applied to interpreting specific real VSP observations as well as explaining the results of physical modelling for a 3-D crack model with the estimation of crack parameters.Key wordsRay tracingkinematic inversionhigh-order perturbationssensitivity functionsanisotropycracksVSP

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.

Prediction of sonic velocities in shales from well logs is important for seismic to log ties if the sonic log is absent for a shaly section, for pore pressure anomaly detection, and for data quality control. An anisotropic differential effective medium (DEM) was used to simulate elastic properties of shales from elastic properties and volume fractions of silt and wet clay (a hypothetical composite material that includes all clay minerals and water). Anisotropic elastic coefficients of the wet clay were assumed as a first-order approximation to be linearly dependent on wet clay porosity (WCP). Here, by WCP we mean a ratio of a pore volume occupied by water to a total volume of the wet clay. Effects of silt inclusions on elastic coefficients of shales were taken into account by using the anisotropic differential effective medium model. Silt inclusions were modeled as spherical quartz particles. Simulated elastic coefficients of shales were used to calculate compressional and shear velocities, and these were in a good agreement with the sonic velocities observed on a test data set from an offshore Mid-Norway well penetrating a 500-m vertical section of shale. To further study the elastic properties of wet clays, elastic coefficients calculated from compressional and sonic velocities measured in shales were inverted for vertical profiles of wet clay elastic coefficients. Analysis of these coefficients found that in the well considered, the increase in elastic coefficients of shales was controlled by the increase of silt fraction with depth. Elastic coefficients of wet clay found no increase with depth. The inverted elastic moduli of wet clay found much stronger correlation with WCP than do the moduli of shale. This confirmed the hypothesis that silt fraction is one of the key parameters for the modeling of elastic properties of shale.

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 dynamic behaviour of a piezoelectric actuator bonded to an anisotropic elastic medium