Formulation of the multiple anisotropic scattering process in two dimensions for anisotropic source radiation

The radiative transfer theory (RTT) describes the energy transport through a random heterogeneous medium, neglecting phase information. It provides an adequate framework for modelling high-frequency seismogram envelopes. For isotropic scattering and sources, the radiative transfer equation (RTE) has been formulated analytically and numerically simulated using Monte Carlo methods for acoustic and elastic media. Here, we derive an exact analytical solution of the RTE in 2-D space for the acoustic case, including anisotropic scattering for a anisotropic point-like impulsive source. For this purpose, we generalize the path integral method, which has been used before in the isotropic case, to take into account the anisotropy of both the source radiation pattern and scattering processes, simultaneously. Then we obtain a general solution, which is written in a closed form in the Fourier space. To illustrate the theoretical results, we compute the full space and time evolution of the specific intensity for an arbitrary case. We also compare the time traces computed from our general solution with cases in which the source and/or the scattering process are isotropic. The importance of taking into account both anisotropies simultaneously becomes obvious in our examples. We also show that at long lapse time, our example approaches the solution of the diffusion equation.     

Seismic waves in stratified anisotropic media

Summary. The response of a structure composed of anisotropic strata can be built up from the reflection and transmission properties of individual interfaces using a slightly modified version of the recursion scheme of Kennett. This scheme is conveniently described in terms of scatterer operators and scatterer products. The effects of a free surface and the introduction of a simple point source at any depth can be accommodated in a manner directly analogous to the treatment for isotropic structures. As in the isotropic case the results so obtained are stable to arbitrary wavenumbers. For isotropic media, synthetic seismograms can be constructed by computing the structure response as a function of frequency and radial wavenumber, then performing the appropriate Fourier and Hankel transforms to obtain the wavefield in time-distance space. Such a scheme is convenient for any system with cylindrical symmétry (including transverse isotropy). Azimuthally anisotropic structures, however, do not display cylindrical symmétry; for these the transverse component of the wavenumber vector will, in general, be non-zero, with the result that phase, group, and energy velocities may all diverge. The problem is then much more conveniently addressed in Cartesian coordinates, with the frequency-wavenumber to time-distance transformation accomplished by 3-D Fourier transform.     

Reflection coefficients for weak anisotropic media

The interaction of plane elastic waves with a plane boundary between two anisotropic elastic half-spaces is investigated. The anisotropy dealt with in this study is of a general type. Explicit expressions for energy-related reflection and transmission coefficients are derived. They represent an approximation which is valid for a small deviation of the elastic parameters from isotropy.
Classical perturbation theory is applied on a 6times6 non-symmetric real eigenvalue problem to calculate first-order corrections for the polarization and stress of the plane waves. The explicit solution of the isotropic problem is used as a reference case. Degenerate perturbation theory is used to consider the splitting of the isotropic S -wave into two anisotropic qS-waves. The boundary conditions for two half-spaces in welded contact lead to a 6times6 system of linear equations. A correction to the isotropic solution is calculated by linearization. The resultant coefficients are functions of horizontal slowness, Lamé parameters and densities of the reference media, and of the perturbation of the elasticity tensors from isotropy.     

Energy flux in viscoelastic anisotropic media

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.