ALBERT

Notes: Maslov ray summation is ‘less local’ than ordinary ray theory, because the receiver waveform depends on non-Fermat or neighbouring rays and more information about the wavefront than just local Gaussian curvature. In this way, the Maslov solution is able to remain valid at caustics, where geometrical rays and corresponding stationary points of the Maslov phase coalesce. the wavefront information is expressed via the Legendre transformation, whereby the physical wavefront is represented as the envelope of a family of tangent ‘planes’ (Snell fronts). the actual form of the Snell fronts (true planes, sections of curves or surfaces, etc.) depends on the spatial coordinates used. Given a selection for the Snell fronts and Maslov phase, one can substitute the Maslov integral solution directly into the wave equation and obtain a transport equation for the Maslov amplitude. This direct substitution is analogous to that used in ordinary ray theory and avoids pseudo-differential operators.Sometimes the relative curvature of the physical wavefront and a tangential Snell front is zero. the envelope-forming process breaks down, because the local correspondence between the physical front and the Snell fronts is not one to one and invertible. This situation corresponds to a so-called ‘pseudo-caustic’ (slowness-domain caustic or telescopic point) in the Maslov solution. Pseudo-caustics are not real. A particular ray from the source may touch a pseudo-caustic at some time in one coordinate system, but in another system this ray will not have a pseudo-caustic (at the same time and place). It is easy to design a change of coordinates (e.g. from cartesian to curvilinear or polar) to deform a single-valued traveltime function appropriately, but a multi-valued or folded wavefront, as at a physical or real caustic, is less simple. Catastrophe theory is concerned with putting multi-valued functions into ‘normal forms’ which do not have psuedo-caustics. the manifold here is ‘Lagrangian’ and V. I. Arnold showed that a special type of deformation or ‘canonical transformation’ must be used. A ‘Lagrangian equivalence’ consists of a deformation of the ‘base’ (x-space) and/or the addition of a function on the base. the latter simply means factoring out an appropriate reference phase before Legendre transformation and we have found that this simple step is often sufficient for removing pseudo-caustics. It requires no new numerical work, only an inspection or understanding of the ray-tracing results at hand.We present some body-wave computations using the reference-phase technique for models with real caustics in 2-D and for a single-valued wavefront in 3-D. We point out that a Lagrangian equivalence may be used to turn a maximum of the Maslov phase function into a minimum. This has no effect on the frequency-domain solution, but may affect the causality of the computed waveform when the Chapman method is used to obtain the time-domain response. Causality is a property which one may need to impose explicitly. Only the non-delta or one-sided function part of the response (waveform tail) is affected by this consideration.Although zeroth-order Maslov theory correctly describes the severe waveform is clear from Secdistortion due to wavefront catastrophes, it may not adequately model the more subtle effects of smooth wavefront bending. Zeroth-order Maslov theory contains some but not all of the first-order (ω−1) terms of ordinary asymptotic ray theory. First-order Maslov theory is needed for complete consistency up to ω−1. Experimentation will several different zeroth-order Maslov representations is a simple, rapid way to ascertain the potential importance of thse more subtle waveform effects. If the waveform tails are too strong, the assumption that the Maslov (and ray theory) amplitude function can be expanded in powers of ω− may break down. Numerical integration of a wave equation is then necessary.

Notes: We describe the effects of anisotropic slowness-surface conical points (acoustic axes) on quasi-shear wavefronts and waveforms in variable elastic media.Conical points have quite complicated geometrical consequences even for a point source in or a wave refracting into a homogeneous anisotropic medium. A hole develops in the fast quasi-shear wavefront and the swallowtail catastrophe plays an important role in the geometry of the slow quasi-shear front, which becomes folded with numerous self-intersections. The two fronts are joined along the rim of the hole. This geometry influences the waveforms, which show Hilbert-transform and diffraction effects. Therefore, standard ray theory is inapplicable even for a uniform medium and the Maslov method is needed to describe waveforms.The introduction of elastic gradients further complicates the geometry of the problem, because rays bend sharply as their slowness approaches that of the axis. An initially smooth, single-valued slow quasi-shear front will evolve in the gradient region into a front which is folded and multivalued and once again the swallowtail is important. However, in contrast to a homogeneous medium, no‘hole’develops in the fast quasi-shear front and the slow and fast fronts separate completely. While such geometrical factors are included in the Maslov method, waveforms are also affected by coupling of the fast and slow waves on nearing the axis, where the rays and polarizations rotate most rapidly and their slownesses differ by very little.Numerical examples are presented for a cubic and an orthorhombic material. The differences between these two examples show that the fine structure of‘continuously varying internal conical refraction’can vary considerably from material to material, though its basic principles are clearly defined. Waveforms are presented for a point source in a uniform medium and for fast and slow shear waves in a gradient, with and without coupling. Overall, we conclude that the wavefront-folding effects cause the most drastic waveform distortions. Coupling becomes most important when signals merge, as at cuspidal edges or at lower frequencies, since the net waveform could be altered significantly if its components are varied. Conical refraction complicates and yet could be decisive for the identification of seismic anisotropy and rock properties.

Notes: Lighthill and others have expressed the ray-theory limit of Green's function for a point source in a homogeneous anisotropic medium in terms of the slowness-surface Gaussian curvature. Using this form we are able to match with ray theory for inhomogeneous media so that the final solution does not depend on arbitrarily chosen ‘ray coordinates’ or ‘ray parameters’ (e.g. take-off angles at the source). The reciprocity property is clearly displayed by this ‘ray-coordinate-free’ solution. The matching can be performed straightforwardly using global Cartesian coordinates. However, the ‘ray-centred’ coordinate system (not to be confused with ‘ray coordinates’) is useful in analysing diffraction problems because it involves 2 times 2 matrices not 3 times 3 matrices. We explore ray-centred coordinates in anisotropic media and show how the usual six characteristic equations for three dimensions can be reduced to four, which in turn can be derived from a new Hamiltonian. The corresponding form of the ray-theory Green's function is obtained. This form is applied in a companion paper.

Notes: We consider the shear-wave polarization vectors and rays as a wavefront passes from an isotropic region into one where the anisotropy increases gradually from zero. This is a necessary first step when analysing the solution of the wave equation in such a transition. The method involves an idealized situation where there is a jump in, say, the third derivative of the elastic parameters along the border between the isotropic and anisotropic regions. Velocity and slowness on the incident rays are then continuous with those on the splitting rays in the anisotropic region. Higher derivatives of velocity and slowness are discontinuous, however, and these jumps are found. The first derivative of polarization wrt time along rays is generally discontinuous, although its component in the ray direction itself is conserved. This discontinuity in the rate of change of polarization is used to demonstrate how, or why, the usual asymptotic ray method fails to satisfy the wave equation in such a transition region. The results are used in a companion paper on the transition zone problem and the methods should be useful in other problems where shear wavespeeds coalesce and the geometry of the rays controls the resulting interference.

Notes: Traveltimes and amplitudes for P-waves from a 500 km-deep source in the Kuril subduction zone have been synthesized by ray tracing in smooth 3-D models that allow general anisotropy and inhomogeneity. the aim is to compare the effects of proposed anisotropy in or near slabs with those of lateral heterogeneity alone. to concentrate on these effects, the source position, slab thickness (90 km), dip (63°) and velocity anomaly (5 per cent) are held constant. Results are presented for isotropic models with slab penetration to 670km and 1000 km. Anisotropic models with 670 km-deep slabs have anisotropy within the slab (Anderson 1987) and in a 10° wedge above the slab (Ribe 1989a; McKenzie 1979). the resulting wavefront topology is never as simple as that in a laterally homogeneous reference Earth and there is strong model dependence of shadow zones, caustics and areas of multipathing.Rays are traced through slab models defined by 3-D cubic-spline interpolation of up to 21 elastic constants. Outside the slab region, 1-D ray tracing through PREM and spherical trigonometry are used to complete the ray path. the results illustrate the importance of using both traveltime and amplitude information when interpreting slab structure from teleseismic data. Some anisotropic slab models have been found which produce large (〉2 s) traveltime residuals that are similar in many parts of the world to those for the deep isotropic model, but the amplitude patterns are substantially different. the model with a deep isotropic slab produces a narrow band of large traveltime residuals (3 s) and high amplitudes in a region across northern Canada. This feature is due to the focusing of rays that have travelled down the high-velocity core of the deep slab. Regions where ray theory fails (i.e. caustics) are obvious through multipathing and amplitude singularities. Hilbert-transforms and Airy-type decay caustics should be observed in many places if the models presented are good representations of the Earth. Multipathing in along-strike regions is a pervasive feature of the models considered and the degree of such multipathing is highly dependent on the nature of the slab-boundary velocity gradients. the model with anisotropy above the slab produces multipathing (traveltime triplication) in the down-dip region (i.e. a narrow region through Europe). Identifying such non-linear or ‘catastrophic’ features in teleseismic data is potentially more diagnostic than linearized interpretations (automated inversion). Overall, the results show that a range of conservative models representing a range of structural theories can encompass a wide range of wavefront consequences.Drs M. Weber and V. Červený are thanked for helpful reviews of the manuscript. Advice and reprints from Dr K. Fujita are appreciated and Dr D. L. Anderson is thanked for suggesting the anisotropic slab model with no isotropic velocity anomaly. the authors acknowledge the support of NSERC (Canada) through Operating Grant number A1465. J-M.K. is supported by an Amoco Postgraduate Scholarship and an Ontario Graduate Scholarship.

Notes: The usual high-frequency (ω) ray method of modelling reflections from smooth boundaries does not account for interference arising near critical angles of incidence. When the reflector is effectively convex towards the incident-wave side, the ray method fails (i) in a transition zone of width O(ω-1/2) separating regions of partial and total ray reflection, (ii) for the transmission in a zone along the reflector having thickness O(ω-1/2) and where a whispering-gallery is formed by multiply-reflected turning waves, and (iii) for turning rays refracted back into the first medium at distances from the critical point less than O(mω-1/4), where m is the number of turns.The interference is a local effect and approximate analytic waveforms can be found for laterally varying media. The 2-D acoustic case is described here. The fields near the point of critical incidence and about the critically reflected ray are found in a way similar to the grazing ray solution described in an earlier paper. However, through the critical ray problem the first two terms of the asymptotic expansion must be considered. The whispering-gallery is obtained by exploiting work on modes by Buldyrev, Lewis, Ludwig and others. The modal solution is then matched with the solution near the critical point to determine the initial modal amplitudes. The individual ray contributions separate from the modes as the waves progress along the boundary. The modal form of solution is needed, even if one wishes to demonstrate only the matching with ray theory for the primary refracted/turning wave near the critical point. This is because the transition zone for the refraction is much wider than that for the reflected wave.Though it is more complicated than the earlier grazing ray solution, the critical-ray diffracted wavefield is still continued way from the boundary using ray coordinates. Hence, it is not difficult to provide ray tracing programs with corrections to ray synthetics. Some preliminary numerical examples are presented to indicate the waveforms and potential significance of these corrections. They are small in comparison with the total reflection, but they are very significant in comparison with the weaker refracted waves. Even where ray theory is (just) valid, for practical reasons one may wish to avoid tracing multiple reflections close to a numerically specified boundary. The boundary-layer formulae provide a more ‘natural’ solution to both the theoretical and numerical problems for the refracted part of the field.

Notes: When a P-wave is incident on an isotropic-anisotropic boundary, the reflected S conversion will generally contain some transverse (SH) component. Numerical results show that the magnitude of this SH component is strongly related to the transmitted qP-wave and the form of the P-wave anisotropy (degree and orientation) in the lower medium, rather than the jump in shear wave velocities over the interface. Varying Poisson's ratio in the incident medium changes the amplitude of the reflected SV component, as one would expect, but has minimal effect on the reflected SH signal. Therefore, it is possible to obtain reflected S-waves that are almost purely transverse, even though the source is compressional and the medium of propagation is isotropic. This study of these indirect effects of anisotropy was prompted by anomalous transverse signals in a refraction data set, which is included for comparison.

Notes: An S-wavefront from an isotropic region is expected to separate into two fronts when it passes into a gradually more anisotropic region. Standard ray expansions may be used to continue the waves in the anisotropic region when these two S-wavefronts have separated sufficiently. However, just inside the anisotropic region the two S-waves interfere with an effect that is stronger than the usual ω-1 corrections of the ray method. A waveform distortion can occur and this should be considered when modelling S-waves in, e.g., subduction zones with regions of isotropy grading into regions of anisotropy.The interference is studied here by local analysis of an integral equation obtained by the Green's function method. It is found that if the elasticity and its first two derivatives are continuous at the isotropy/anisotropy border, then zeroth-order ray theory may still be used to continue the incident wave into the anisotropic region. The incident displacement is simply resolved into two definite directions at the point where the anisotropy begins. These two directions are the limits of the unique eigenvectors on the anisotropic rays as the point of isotropy (onset of splitting) is approached. If the nth derivative of the elasticity is discontinuous at the isotropy/anisotropy border, then the scattering integral which describes the interference makes a correction to ray theory which is O(ω-1/n+1) in magnitude. Hence, the interference effect is stronger when the emergence of anisotropy is more gradual.Although the corrections are given by simple expressions, it is not reasonable to specify numerical velocity models up to such high-order derivatives. For a smooth interpolation scheme, such as cubic splines, it is more practical to monitor the splitting rays obtained by ray tracing and to use the best-fitting ‘equivalent’ high-order discontinuity. This will lead to an estimate of the importance of the correction terms. An example is given for a subduction zone model involving olivine alignment in the mantle-wedge above the slab.

Notes: Germination of annual pasture species was studied under controlled-environment conditions in south-western Australia at temperatures in the range from 4°C to 35°C. Subterranean clover (Trifolium subterraneum) and Wimmera ryegrass (Lolium rigidum) had a germination of 90% between 12°C and 29°C, whereas capeweed (Arctotheca calendula) had a high germination percentage in a much narrower temperature range with an optimum of 25°C. Growth of subterranean clover, capeweed and Wimmera ryegrass between 28 and 49 days after sowing (DAS) was also studied at two photon flux densities, 13 and 30 mol m−2 d−1, and at diel temperatures in the range from 15/10°C to 33/28°C. Pasture species grown at a density of 1000 plants m−2 accumulated at least twice the amount of shoot dry matter when subjected to temperatures of 21/16°C and 27/22°C, compared with a lower temperature of 15/10°C and a higher temperature of 33/28°C. Except at the highest temperature and at high photon flux density, capeweed had lower green area indices (GAI) than the other two species at 28 DAS. Crop growth rates between 28 and 49 DAS were higher in Wimmera ryegrass than in the other two species, whereas subterranean clover had a lower relative growth rate than the other two species at all temperatures and both photon flux densities. Subterranean clover and capeweed intercepted a greater proportion of the incident radiation compared with Wimmera ryegrass. The values of radiation interception and GAI were used to estimate the number of DAS to reach 75% radiation interception [f(0·75)]. The number of days to reach f(0·75) decreased with increasing temperature from 15/10°C to reach a minimum at 27/22°C. The time taken to achieve f(0·75) was always shorter by about 10 d when the photon flux density was 30 mol m−2 d−1 in the autumn compared with 13 mol m−2 d−1 in the winter. These results are discussed in relation to the early growth of annual pasture in the field.

Notes: We point out that while the equations of seismic ray geometrical spreading given recently by Norris do differ as stated from equations given by Červeny, it does not imply that the latter equations are wrong. The two sets of equations differ only in form and in a way which, in part at least, can be ascribed to different choices of Hamiltonian by the two authors. Quantitatively, though, the two sets of equations are entirely equivalent. We also present some numerical results of ray tracing in anisotropic models simulating a continential rift, a spreading ridge and a subduction zone. These three structures span a range of geological mechanisms for seismic anisotropy. Though definitive conclusions cannot be easily drawn when there is both anisotropy and inhomogeneity, the results do indicate the magnitude of ray path, travel-time and amplitude variations to be expected for P-waves when anisotropy is introduced.