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.

Notes: In principle, crosshole, traveltime tomography is ideal for directly detecting and measuring seismic anisotropy. The traveltimes of multiple rays with wide angular coverage will be sensitive both to inhomogeneities and to anisotropy. In practice, the traveltimes will depend only on a limited number of the anisotropic velocity parameters, and the data may not be adequate even to determine these parameters uniquely. In addition, trade-offs may exist between anisotropy and inhomogeneities. In this paper, we use the linear perturbation theory for traveltimes in general, weakly anisotropic media to discuss the dependence of traveltimes in 2-D crosshole tomographic experiments on the anisotropic parameters. In a companion paper, we apply the results to synthetic and real data examples. We show that when measurements are restricted to a 2-D plane, the qP and qS traveltimes depend on subsets of the complete set of 21 anisotropic velocity parameters. Formulae are developed for the differential coefficients of the traveltimes with respect to these parameters in piecewise homogeneous and in linearly interpolated models. It is shown how in a generally oriented model element, the local parameters are related to the same parameters in the global model. The parameters that can be determined from 2-D tomographic data do not in general determine the full nature of the anisotropy. Rather, these parameters serve only to describe the intersection of the slowness sheet with the 2-D plane. Since many models may fit this description, additional information on symmetry properties and orientations is required. For example, if a priori information suggests that the anisotropy is transversely isotropic (TI), then we can determine some of the TI parameters and some information on the orientation of the axis of symmetry. Formulae are given relating the general parameters to those of a TI system with general orientation of the symmetry axis. The general formulae for qS traveltimes are intrinsically more complicated than those for qP. In the qS case, the traveltime perturbation depends on the polarization, which in turn depends on the perturbation. This makes the general problem non-linear even for small perturbations. However, the mean qS traveltime and the traveltime dependence on various subsets of parameters are linear. Although linear perturbation theory is invalid for qS rays, degenerate perturbation theory is valid for the calculation of the traveltimes and could be used in a non-linear inversion scheme.

Notes: Cross-borehole seismic data have traditionally been analysed by inverting the arrival times for velocity structure (traveltime tomography). The presence of anisotropy requires that tomographic methods be generalized to account for anisotropy. This generalization allows geological structure to be correctly imaged and allows the anisotropy to be evaluated. In a companion paper we developed linear systems for 2-D traveltime tomography in anisotropic media. In this paper we analyse the properties of the linear system for quasi-compressional waves and invert both synthetic and real data. Solutions to the linear systems consist of estimates of the spatial distributions of five parameters, each corresponding to a linear combination of a small subset of the 21 elastic, anisotropic velocity parameters. The parameters describe the arrival times in the presence of weak anisotropy with arbitrary symmetries. However, these parameters do not, in general, describe the full nature of the anisotropy. The parameters must be further interpreted using additional information on the symmetry system. In the examples in this paper we assume transverse isotropy (TI) in order to interpret our inversions, but it should be noted that this final interpretation could be reformulated in more general terms.The singular value decomposition of the linear system for traveltime tomography in anisotropic media reveals the (expected) ill-conditioning of these systems. As in isotropic tomography, ill-conditioning arises due to the limited directional coverage that can be achieved when sources and receivers are located in vertical boreholes. In contrast to isotropic tomography, the scalelength of the parametrization controls the nature of the parameter space eigenvectors: with a coarse grid all five parameters are required to model the data; with a fine grid some of the parameters appear only in the null space.The linear systems must be regularized using external, a priori information. An important regularization is the expectation that the elastic properties vary smoothly (an ad hoc recognition of the insensitivity of the arrival times to the fine-grained properties of the medium). The expectation of smoothness is incorporated by using a regularization matrix that penalizes rough solutions using finite difference penalty terms. The roughness penalty sufficiently constrains the solutions to allow the smooth eigenvectors in the null space of the unconstrained problem to contribute to the solutions. Hence, the spatial distribution of all five parameters is recovered. The level of regularization required is difficult to estimate; we advocate the analysis of a suite of solutions. Plots of the solution roughness against the data residuals can be used to find ‘knee points’, but for the fine tuning of the regularization one has little recourse but to examine a suite of images and use geological plausibility as an additional criterion.The application of the regularized numerical scheme to the synthetic data reveals that the roughness penalty should include terms that penalize high gradients addition to penalizing high second derivatives. Only when this constraint was included were the features of the original model recovered. The inversions of the field data yield good images of the expected stratigraphy and confirm previous estimates of the magnitude of the anisotropy and the orientation of the symmetry axis. The solutions further indicate an increase in anisotropy from the top to the bottom of the survey region that was not previously detected.

Notes: It is well known that when a seismic wave propagates through an elastic medium with gradients in the parameters which describe it (e.g. slowness and density), energy is scattered from the incident wave generating low-frequency partial reflections. Many approximate solutions to the wave equation, e.g. geometrical ray theory (GRT), Maslov theory and Gaussian beams, do not model these signals. The problem of describing partial reflections in 1-D media has been extensively studied in the seismic literature and considerable progress has been made using iterative techniques based on WKBJ, Airy or Langer type ansätze. In this paper we derive a first-order scattering formalism to describe partial reflections in 3-D media. The correction term describing the scattered energy is developed as a volume integral over terms dependent upon the first spatial derivatives (gradients) of the parameters describing the medium and the solution. The relationship we derive could, in principle, be used as the basis for an iterative scheme but the computational expense, particularly for elastic media, will usually prohibit this approach. The result we obtain is closely related to the usual Born approximation, but differs in that the scattering term is not derived from a perturbation to a background model, but rather from the error in an approximate Green's function. We examine analytically the relationship between the results produced by the new formalism and the usual Born approximation for a medium which has no long-wavelength heterogeneities. We show that in such a case the two methods agree approximately as expected, but that in a media with heterogeneities of all wavelengths the new gradient scattering formalism is superior. We establish analytically the connection between the formalism developed here and the iterative approach based on the WKBJ solution which has been used previously in 1-D media. Numerical examples are shown to illustrate the examples discussed.

Notes: Ray perturbation theory and the Born approximation have both been used extensively in seismological studies to describe the effects of a slowness perturbation on body and surface wavefields. The relationship between the expressions for the perturbed wavefield calculated using the two methods is investigated here. Using the symplectic symmetry of the ray equations we demonstrate the agreement, in the far field, of the two methods to first order in the slowness perturbation and to leading order in the asymptotic ray series. Thus it is shown that geometrical ray effects, like the traveltime perturbation, ray bending and focusing, are contained within the Born scattering formalism, provided these effects are small. The propagator formalism used to present the results is sufficiently general to include body and surface waves with a smoothly varying inhomogeneous elastic reference medium.