Andrzej Hanyga

The kinematic inverse problem for weakly laterally inhomogeneous anisotropic media

Abstract The analysis of the kinematic inverse problem in anisotropic elastic media displays the indeterminacy of such problems in the presence of anisotropy. Some quantitative information about the elastic moduli must be available a priori. In view of the perturbational treatment applied here it is also assumed that elastic moduli are known in the zero approximation. The objective is to find corrections to elastic moduli corresponding to the observed deviations of travel time data from the zero approximation. If the deviations of travel times as well as the required supplementary information are available, then the kinematic inverse problem can be solved numerically. Allowance is made for weak lateral inhomogeneities. The method presented here applies to the layers lying above the first low velocity layer below the Earths surface.

Dynamic ray tracing in an anisotropic medium

Abstract We derive a dynamic ray tracing system for seismic waves in an anisotropic medium. It consists of ten equations and can be solved numerically. For isotropic media it does not reduce to the DRT systems derived by other authors.

Diffraction of pulses in the vicinity of simple caustics and caustic cusps

Abstract The objective of this paper is an extension of the Geometrical Theory of Diffraction (GTD) taking account of caustics. A closed list of those cuspoid caustics which are stable in the absence of symmetry constraints in a space of at most 5 dimensions is given along with integral expressions for the associated pulse shapes. Ludwigs formulae for harmonic fields are rederived by a new method. Universally valid time-domain expressions are associated with each caustic type. For the simple caustic and caustic cusp time-domain expressions are obtained in closed form. Finally, algorithms for the computation of time-domain expressions for a given wave field based on real and complex ray tracing are discussed.

Spatially fractional-order viscoelasticity, non-locality, and a new kind of anisotropy

A class of non-local viscoelastic equations of motion including equations of fractional order with respect to the spatial variables is studied. It is shown that space-fractional equations of motion of an order strictly less than 2 allow for a new kind of anisotropy, associated with azimuthal dependence of non-local interactions between stress and strain at different material points. Constitutive equations of such viscoelastic media are determined. Relaxation effects are additionally accounted for by replacing second-order time derivatives by lower-order fractional derivatives. Explicit fundamental solutions of the Cauchy problem for scalar equations with isotropic and anisotropic non-locality are constructed. For some particular choices of the parameters, numerical solutions are constructed.

Relaxation, dispersion, attenuation, and finite propagation speed in viscoelastic media

Dispersion and attenuation functions in a linear viscoelastic medium with a positive relaxation spectrum are given by integral representations in terms of a positive Radon measure satisfying a growth condition. Kramers–Kronig dispersion relations with one subtraction can be derived from the integral representations of the dispersion and attenuation functions. The dispersion and attenuation functions have sublinear growth in the high frequency range. The wave number vector can have a linear component in addition to the dispersion function. In this case the viscoelastic waves propagate with a bounded speed. In the other cases viscoelastic wave propagation has a diffusion-like character.

Existence theorems for linear hyperbolic systems with unbounded coefficients

RiassuntoAlcuni teoremi di esistenza vengono stabiliti per il problema (1), (2), (3) quando i coefficientiAij(x, t) sono non limitati inx e discontinui int.SummaryWe shall establish existence theorems for the initial-boundary value problem (1), (2), (3), when the coefficientsAij (x, t) are unbounded inx and discontinuous int.