ALBERT

Notes: The head-wave contribution to a reflection is investigated by two different methods and it is shown that the new result presented by Lerche and Hill [J. Math. Phys. 26, 1420 (1985)] for the head-wave amplitude is in error due to the use of an inappropriate mathematical method.

Notes: Various exact methods of inverting the complete waveform of vertical seismic reflection data to produce acoustic impedance profiles have been suggested. These inverse methods generally remain valid for nonvertical, plane-wave data, provided total reflection does not occur. Thus, in principle, the “seismogram” at each ray parameter in a slant stack can be interpreted separately.Rather than invert each plane-wave seismogram separately, they can all be interpreted simultaneously and an “average” model thus obtained. Inversion for both the velocity and the density also becomes possible when two or more plane-wave seismograms are simultaneously inverted. The theory for a noniterative inversion method, based on the time-domain Riccati equation, is discussed. Numerical examples of inversions using this technique on synthetic data demonstrate its numerical stability and the advantage of simultaneous inversion of several seismograms to reduce the effect of noise in the data and increase the stability of the inversion process.

Notes: Ray tracing through gradients in anisotropic materials is complicated by singularities where the two quasi-shear wave slowness sheets cross or touch. Difficulties associated with such points can be removed by explicitly including polarization in the ray tracing equations. Slowness sheet and wavefront plots show the polarization and velocity behavior of various anisotropy models of aligned cracks in the upper crust. A simple scaling of the elastic tensor with depth can be shown to be approximately correct for models of aligned cracks within an isotropic host matrix with a linear velocity gradient. Ray tracing examples for models of aligned cracks within a strong vertical velocity gradient in the uppermost crust demonstrate various features of azimuthal anisotropy, including amplitude and polarization anomalies and shear-wave splitting. Quasi-shear wave polarizations typically twist along ray paths, with stronger twisting near the symmetry axis in hexagonally symmetric media. Strong anisotropy can cause unusual effects, such as ray paths which have three turning points in laterally homogeneous models.

Notes: Quasi-shear wave polarizations typically twist along ray paths through gradient regions in anisotropic media, causing frequency dependent coupling between the qS-waves. This coupling is much stronger than the analogous coupling between P- and SV-waves in isotropic gradients because of the small difference between the qS-wave velocities. Geometrical ray theory is typically valid for qS-waves only at relatively high frequencies, and does not converge to the isotropic result in the limit of infinitely weak anisotropy. Using the plane-wave layered response, we show examples of this coupling and how it may cause frequency dependent shear-wave polarizations. We consider two special cases where the coupling is especially strong in hexagonally symmetric media: (i) intersection singularities where the slowness sheets cross, and (ii) kiss singularities where the slowness sheets touch at the symmetry axis. We show numerical and asymptotic solutions for the pulses generated in these situations. In some cases, far-field excitation of both quasi-shear waves (and shear-wave splitting) will result from an incident wave composed of only one of the quasi-shear waves.

Notes: Zeroth-order ray theory is frequently used to calculate synthetic seismograms in media which are both anisotropic and inhomogeneous. One of the principal features of such media is that the polarization vectors of the two quasi-shear (qS) waves are determined by the nature of the anisotropy. Thus, a shear wave entering a region of anisotropy will generally be split into two separate polarizations. Ray theory predicts that these two waves will propagate independently, at different velocities, throughout the anisotropic region. Ray theory solutions also show that in inhomogeneous media, the polarization vectors will rotate along the ray. The rotations of these polarization vectors are strongly influenced by the symmetry and orientation of the anisotropy system, but only weakly depend upon the strength of the anisotropy. In contrast, in isotropic media the polarization of S-waves is determined from the initial conditions and only varies slowly due to the ray curvature. The polarization only changes in the ray direction and at any point does not rotate about the ray.In this paper we show that in the limit of infinitely weak anisotropy, solutions calculated using ray theory in anisotropic media conflict with the known results calculated for a similar isotropic medium. We show this fundamental breakdown in ray theory occurs because coupling between the qS waves is ignored in the zeroth approximation. Thus, the isotropic limit is not equivalent to the high-frequency limit of anisotropic ray theory. The coupling is particularly important in weakly anisotropic media, where the qS velocities are similar, but the same effect is still present in media exhibiting stronger anisotropy. This coupling must be taken into account when calculating waveforms.We show that this coupling may be modelled by treating the ‘error’ terms, produced by substituting a zeroth-order ray theory Green's function into the wave equation, as source terms distributed throughout the medium. For weakly anisotropic media where the qS ray paths are similar, this volume integral may be simplified using perturbation and asymptotic methods and evaluated as a simple integral along the ray path. In the isotropic limit this expression correctly describes the polarization of shear waves along the ray. This integral is easy to compute, requiring only quantities already used in ray tracing and traveltime calculations. A prior knowledge of the location, or even the existence of kiss, intersection, point or other singularities along the ray path, is not required for the method to give accurate results. We present some numerical examples for some simple cases previously investigated by less general or more expensive techniques.

Notes: Reciprocal relationships between the plane-wave reflection/transmission coefficients in anisotropic media are derived directly from the transformed wave equations without use of Betti's theorem. If the eigensolutions are normalized correctly, coefficients with the rǒles of the incident and generated waves reversed are equal, provided the sign of slowness parallel to the interface is also reversed.