Fermat's principle for seismic rays in elastic media

We prove that rays in linearly elastic anisotropic nonuniform media obey Fermat's principle of stationary traveltime. First, we formulate the concept of rays, which emerges from the Hamilton equations. Then, we show that these rays are solutions of the variational problem stated by Fermat's principle. This proof is valid for all rays except the ones associated with infection points on the phase-slowness surface.     

The Homogeneous Finite-Difference Formulation of the P-SV-Wave Equation of Motion

Two different approaches to finite-difference modeling of the elastodynamic equations have been used: the heterogeneous and the homogeneous. In the heterogeneous approach, boundary conditions at interfaces are treated implicitly; in the homogeneous, they are explicitly discretized. We present a homogeneous finite-difference scheme for the 2-D P-SV-wave case. This scheme represents a generalization of earlier such schemes, being able to model media with arbitrary non-uniformities, provided only that all interfaces are aligned with the numerical grid. We perform a detailed comparison of the generalized homogeneous scheme with the analogous heterogeneous scheme, and show the two schemes to be identical for media with a spatially constynt Poisson's ratio. For media where Poisson's ratio is spatially varying, the schemes differ by terms first-order in the spatial step size. However, a comparison of the numerical results produced by the two schemes shows that the resulting differences are negligible for a wide range of values of the Poisson's ratio contrast.     

On Choosing Effective Elasticity Tensors Using a Monte-Carlo Method

A generally anisotropic elasticity tensor can be related to its closest counterparts in various symmetry classes. We refer to these counterparts as effective tensors in these classes. In finding effective tensors, we do not assume a priori orientations of their symmetry planes and axes. Knowledge of orientations of Hookean solids allows us to infer properties of materials represented by these solids. Obtaining orientations and parameter values of effective tensors is a highly nonlinear process involving finding absolute minima for orthogonal projections under all three-dimensional rotations. Given the standard deviations of the components of a generally anisotropic tensor, we examine the influence of measurement errors on the properties of effective tensors. We use a global optimization method to generate thousands of realizations of a generally anisotropic tensor, subject to errors. Using this optimization, we perform a Monte Carlo analysis of distances between that tensor and its counterparts in different symmetry classes, as well as of their orientations and elasticity parameters.     

On Christoffel roots for nondetached slowness surfaces

The only restriction on the values of the elasticity parameters is the stability condition. Within this condition, we examine the Christoffel equation for nondetached qP slowness surfaces in transversely isotropic media. If the qP slowness surface is detached, each root of the solubility condition corresponds to a distinct smooth wavefront. If the qP slowness surface is nondetached, the roots are elliptical but do not correspond to distinct wavefronts; also, the qP and qSV slowness surfaces are not smooth.     

Modelling the potential transport of tropical fish larvae in the Leeuwin Current

Application of an individual-based particle tracking model to the migration of tropical fish larvae along the continental shelf between the Houtman Abrolhos Islands and Rottnest Island (Western Australia) has shown that there is potential for the southwards advection of passive particles/larvae in the Leeuwin Current system throughout the year. However, seasonal variations in the prevailing wind field result in corresponding seasonal changes in the surface current flow (both alongshore and cross-shelf) on the continental shelf, leading to a pulse of modelled particles arriving at Rottnest Island during the autumn months. This matches, within a month, the observed April/May peak in annual recruitment of 2 species of damselfish (Abudefduf sexfasciatus and A. vaigiensis), at the time when the Leeuwin Current is strengthening. It is assumed that the larvae are in the uppermost 20 m of the water column and that there are no vertical diurnal movements.The model has a 10 km grid spacing, and so can resolve some of the current gradients across the continental shelf. Comparison of the modelled near-surface currents with ADCP measurements at 2 current mooring sites as well as with a broader range of historical current measurements off south-western Australia indicates that the alongshore net transport is reasonably well reproduced by the model, whereas agreement with the cross-shelf flow is not as good (this may be partly attributed to the paucity of high-quality near-surface current measurements in the area of study). Because of limitations in our knowledge of the swimming ability, choice of swimming direction and habitat selection of larval fish, as well as the inability of the model to reproduce the small-scale circulation around Rottnest Island, the swimming capacity of the late-stage larval fish is not specifically included; they are considered as potential settlers once they have reached within 20 km of the Island.The observed inter-annual variability in recruitment is not as well matched by the model as is the seasonal pattern, but this is almost certainly due to uncertainties in biological factors such as spawning strength, food and predation en route, which are not known.The modelled results are also applied to a more general discussion of the transport of eggs and larvae of commercial fish and invertebrate species on the Western Australian continental shelf, and it is shown that the seasonality and position on the shelf of the spawning may play a large role in the movement (and hence survival and ultimate recruitment) of different species.     

Classes of Anisotropic Media: A Tutorial

The purpose of the present article is to give a precise definition and analysis from first principals of anisotropy, as the term applies to elastic media, taking care to avoid unnecessary assumptions. Two fundamental concepts, material invariance and symmetry group of a material, are defined purely in terms of the stress-strain relation. The implications of material symmetry, or in other words, of anisotropy, for the structure of the stiffness tensor are then investigated. Using the reduced notation of Voigt, these results are presented as the well-known simplifications in the form taken by the six-by-six stiffness matrix that represents the material's stiffness tensor. A new, simple proof is given for the remarkable fact that an elastic medium cannot have rotational symmetry by an angle of less than 90° without being transversely isotropic. In addition, the mutual relation that the notions of elastic symmetry and crystal symmetry have with respect to the so-called orthogonal group is sketched. Despite the historical association between anisotropic elastic materials and the study of crystals, the given presentation shows that conceptually the notion of anisotropy in elastic media is entirely independent of that of crystal symmetry.     

VSP traveltime inversion for anisotropy in a buried layer

We present a method for calculating the anisotropy parameter of a buried layer by inverting the total traveltimes of direct arrivals travelling from a surface source to a well‐bore receiver in a vertical seismic profiling (VSP) geometry. The method assumes two‐dimensional media. The medium above the layer of interest (and separated from it by a horizontal interface) can exhibit both anisotropy and inhomogeneity. Both the depth of the interface as well as the velocity field of the overburden are assumed to be known. We assume the layer of interest to be homogeneous and elliptically anisotropic, with the anisotropy described by a single parameter χ. We solve the function describing the traveltime between source and receiver explicitly for χ. The solution is expressed in terms of known quantities, such as the source and receiver locations, and in terms of quantities expressed as functions of the single argument x_r, which is the horizontal coordinate of the refraction point on the interface. In view of Fermat's principle, the measured traveltime T possesses a stationary value or, considering direct arrivals, a minimum value, . This gives rise to a key result ‐‐ the condition that the actual anisotropy parameter . Owing to the explicit expression , this result allows a direct calculation of in the layer of interest. We perform an error analysis and show this inverse method to be stable. In particular, for horizontally layered media, a traveltime error of one millisecond results in a typical error of about 20% in the anisotropy parameter. This is almost one order of magnitude less than the error inherent in the slowness method, which uses a similar set of experimental data. We conclude by detailing possible extensions to non‐elliptical anisotropy and a non‐planar interface.