ALBERT

Notes: [Auszug] The rifting of continents and evolution of ocean basins is a fundamental component of plate tectonics, yet the process of continental break-up remains controversial. Plate driving forces have been estimated to be as much as an order of magnitude smaller than those required to rupture thick ...

Notes: [Auszug] FIG. 1 A sketch of the seismic S-wave phases analysed at epicentral distances 75° and 90°. The 660-km discontinuity separates the upper and lower mantle; LIMA refers to the region of upper-mantle anisotropy. Like those analysed in this study, the earthquake (indicated by an asterisk) is ...

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: 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: 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: 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: 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.