Attenuation vector in heterogeneous, weakly dissipative, anisotropic media

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

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.

Gaussian beam migration (GBM), as it is implemented today, efficiently handles isotropic inhomogeneous media. The approach is based on the solution of the wave equation in ray‐centered coordinates. Here, I extend the method to work for 2-D migration in generally anisotropic inhomogeneous media. Extension of the Gaussian‐beam method from isotropic to anisotropic media involves modification of the kinematics and dynamics in the required ray tracing. While the accuracy of the paraxial expansion for anisotropic media is comparable to that for isotropic media, ray tracing in anisotropic media is much slower than that in isotropic media. However, because ray tracing is just a small portion of the computation in GBM, the increased computational effort in general anisotropic GBM is typically only about 40%. Application of this method to synthetic examples shows successful migration in inhomogeneous, transversely isotropic media for reflector dips up to and beyond 90°. Further applications to synthetic data of layered anisotropic media show the importance of applying the proper smoothing to the velocity field used in the migration. Also, tests with synthetic data show that the quality of anisotropic migration of steep events in a medium with velocity increasing with depth is much more sensitive to the Thomsen anisotropy parameter ε than to the parameter δ. Thus, a good estimate of ε is needed to apply anisotropic migration with confidence.

Nearly all dip‐moveout correction (DMO) implementations to date assume isotropic homogeneous media. Usually, this has been acceptable considering the tremendous cost savings of homogeneous isotropic DMO and considering the difficulty of obtaining the anisotropy parameters required for effective implementation. In the presence of typical anisotropy, however, ignoring the anisotropy can yield inadequate results. Since anisotropy may introduce large deviations from hyperbolic moveout, accurate transformation to zero‐offset in anisotropic media should address such nonhyperbolic moveout behavior of reflections. Artley and Hale’s v(z) ray‐tracing‐based DMO, developed for isotropic media, provides an attractive approach to treating such problems. By using a ray‐tracing procedure crafted for anisotropic media, I modify some aspects of their DMO so that it can work for v(z) anisotropic media. DMO impulse responses in typical transversely isotropic (TI) models (such as those associated with shales) deviate substantially from the familiar elliptical shape associated with responses in homogeneous isotropic media (to the extent that triplications arise even where the medium is homogeneous). Such deviations can exceed those caused by vertical inhomogeneity, thus emphasizing the importance of taking anisotropy into account in DMO processing. For isotropic or elliptically anisotropic media, the impulse response is an ellipse; but as the key anisotropy parameter η varies, the shape of the response differs substantially from elliptical. For typical η > 0, the impulse response in TI media tends to broaden compared to the response in an isotropic homogeneous medium, a behavior opposite to that encountered in typical v(z) isotropic media, where the response tends to be squeezed. Furthermore, the amplitude distribution along the DMO operator differs significantly from that for isotropic media. Application of this anisotropic DMO to data from offshore Africa resulted in a considerably better alignment of reflections from horizontal and dipping reflectors in common‐midpoint gather than that obtained using an isotropic DMO. Even the presence of vertical inhomogeneity in this medium could not eliminate the importance of considering the shale‐induced anisotropy.

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

Rocks are far from being isotropic and elastic. Such simplifications in modeling the seismic response of real geological structures may lead to misinterpretations, or even worse, to overlooking useful information. It is useless to develop highly accurate modeling algorithms or to naively use amplitude information in inversion processes if the stress-strain relations are based on simplified rheologies. Thus, an accurate description of wave propagation requires a rheology that accounts for the anisotropic and anelastic behavior of rocks.This work presents a new constitutive relation and the corresponding time-domain wave equation to model wave propagation in inhomogeneous anisotropic and dissipative media. The rheological equation includes the generalized Hooke's law and Boltzmann's superposition principle to account for anelasticity. The attenuation properties in different directions, associated with the principal axes of the medium, are controlled by four relaxation functions of viscoelastic type. A dissipation model that is consistent with rock properties is the general standard linear solid. This is based on a spectrum of relaxation mechanisms and is suitable for wavefield calculations in the time domain.One relaxation function describes the anelastic properties of the quasi-dilatational mode and the other three model the anelastic properties of the shear modes. The convolutional relations are avoided by introducing memory variables, six for each dissipation mechanism in the 3-D case, two for the generalized SH-wave equation, and three for the qP - qSV wave equation. Two-dimensional wave equations apply to monoclinic and higher symmetries.A plane analysis derives expressions for the phase velocity, slowness, attenuation factor, quality factor and energy velocity (wavefront) for homogeneous viscoelastic waves. The analysis shows that the directional properties of the attenuation strongly depend on the values of the elasticities. In addition, the displacement formulation of the 3-D wave equation is solved in the time domain by a spectral technique based on the Fourier method. The examples show simulations in a transversely-isotropic clayshale and phenolic (orthorhombic symmetry).

With the use of the exact expression for the local propagation matrix that was obtained by Wöhler et al. [ J. Opt. Soc. Am. A5, 1554 ( 1988)], we show explicitly that the Berreman 4 × 4 propagation matrix method for layered inhomogeneous anisotropic media reduces to the Abelés 2 × 2 propagation matrix method for layered inhomogeneous isotropic media when the anisotropic medium becomes isotropic. The optical properties of isotropic and anisotropic media have been extensively studied, and a number of schemes for the computation of the optical effects of layered inhomogeneous media have been proposed. Propagation matrix methods have been shown to be particularly useful for studying the reflection and the transmission of electromagnetic waves by layered inhomogeneous isotropic and anisotropic media.1–5 Generally, the Abelés 2 × 2 matrix method is used for layered inhomogeneous isotropic media,1,2,4,5 and the Berreman 4 × 4 matrix method is used for layered inhomogeneous anisotropic media.3 The 4 × 4 matrix method is more general than the 2 × 2 matrix method and is applicable to problems involving media of low optical symmetry.

Formulae for the zeroth-order principal term plus the first-order additional term of the qP- and qS-wave Green's functions in the so-called quasi-isotropic (QI) approximation are derived for an unbounded inhomogeneous weakly anisotropic medium. The basic idea of this approximation is to seek the asymptotic solution of the elastodynamic equation as an expansion with respect to two small parameters of the same order: the small parameter used in the standard ray method and a parameter characterizing differences of tensors of elastic parameters of a weakly anisotropic medium and of a nearby 'background' isotropic medium. As a result, the procedure of constructing the Green's functions is split into two steps: (1) calculation of rays, traveltimes, the geometrical spreading and polarization vectors in the background isotropic medium; (2) calculation of corrections of traveltimes, amplitudes and polarization vectors due to the deviation of the weakly anisotropic medium from the isotropic background.

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.

On the viscoelastic-electromagnetic-gravitational analogy

The mathematical description of phenomena related to wave propagation in anisotropic media is considerably more complicated than that for isotropic media. This complexity stems from many physical properties distinguishing the anisotropic from isotropic medium. Therefore, calculating the angles of reflection and transmission for a ray impining on a boundary between two anisotropic media become more complicated in mathematical implication. Snell law in isotropic medium has not been applicable in anisotropic medium, which must be modified. This article presents a new method of calculating the angles of reflection and transmission on a boundary between two anisotropic media, called as the modified Snell law.

Performance of convolutional perfectly matched layers for pseudospectral time domain poroviscoelastic schemes

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.

Reverse time migration (RTM) exhibits great advantages over other imaging methods because it is based on computing numerical solutions to a two-way wave equation. It does not suffer from dip limitation like one-way downward continuation techniques do, thus enabling overturned reflections to be imaged. As well as correctly handling multipathing, RTM has the potential to image internal multiples when the boundaries responsible for generating the multiples are present in the model. In isotropic media, one can use a scalar acoustic wave equation for RTM of pressure data. In anisotropic media, P- and SV-waves are coupled together so, formally, elastic wave equations must be used for RTM. A new wave equation for P-waves is proposed in tilted transversely isotropic (TTI) media that can be solved as part of an acoustic anisotropic RTM algorithm, using standard explicit finite differencing. If the shear velocity along the axis of symmetry is set to zero, stable numerical solutions can be computed for media with a vertical axis of symmetry and ε not less than δ. In TTI media with rapid variations in the direction of the axis of symmetry, setting the shear velocity along the axis of symmetry to zero can cause numerical solutions to become unstable. A solution to this problem is proposed that involves using a small amount of nonzero shear velocity. The amount of shear velocity added is chosen to remove triplications from the SV wavefront and to minimize the anisotropic term of the SV reflection coefficient. We show modeling and high-quality RTM results in complex TTI media using this equation.

Anisotropic and isotropic solvent effects on the dipole moment and photophysical properties of rhodamine dyes