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SUMMARY In situ stress monitoring of crustal rocks is desirable as it yields insights into earthquake mechanisms, volcanic eruptions and changes in hydrocarbon reservoirs. Shear wave splitting, induced by stress-controlled cracks in the shallow crust, provides a way to infer this stress. Temporal variations in these observations can be difficult to quantify due to scatter in the data and discontinuous observations. Here we present evidence of temporal variations in shear wave splitting from a continuous time-series with a high occurrence of microseismic events recorded in a borehole. We interpret these observations in terms of stress-controlled cracks and are able to infer changes in stress and, via modelling, suggest the cause of the anisotropy. Possible origins of the temporal variation in per cent anisotropy are tidal or oil-field production processes. Our results suggest that shear wave splitting is a viable probe for inferring changes in crustal stress in cracked rock.

When synthetic seismograms are computed for complicated velocity models using asymptotic methods based on ray theory, the caustic and/or pseudo-caustic geometrical catastrophes that often occur in the ray manifolds can cause many difficulties, because ray amplitudes can become infinite, inaccurate or vanish in such situations. The longer the propagation path relative to the scale of the velocity inhomogeneities and the more the propagation tends to be oblique rather than normal to velocity variations, the more troublesome the catastrophes become. Although the Maslov integral method (MI) largely cures the problems of the classical geometric ray theory (GRT) at caustics (where nearby rays in geometrical space cross) and in geometric shadow zones, it suffers from serious artefacts at pseudo-caustics (the crossing points of rays drawn in the mixed geometrical and slowness space of the Maslov integral). Some methods have been devised to overcome this limitation, but none is satisfactory when it is necessary to compute automatically large suites of seismograms for complex models without continuous intervention from a skilled seismologist. In this paper, we demonstrate how combining the Maslov integral method with a stage of Kirchhoff integration can greatly reduce MI's pseudocaustic problems while retaining MI's ability to overcome the caustic and shadow problems of GRT. In many cases, the additional computational burden increases by one order of magnitude but is still much less than that required in a finite-difference modelling, and is not great for a common workstation. The Maslov–Kirchhoff (MK) technique is shown to be substationally more accurate than ordinary Maslov integration when pseudo-caustics are present. Furthermore, it is more robust and automated than the phase-partitioning technique in curing the problem of joint caustics and pseudo-caustics. We also show that the MK scheme can be more efficient than other Kirchhoff modelling techniques where the kernel of integration is obtained by GRT.

SUMMARY The Maslov asymptotic method addresses some of the problems with standard ray theory, such as caustics and shadows. However, it has been applied relatively little, perhaps because its accuracy remains untested. In this study we examine Maslov integral (MI) seismograms by comparing them with finite-diVerence (FD) seismograms for several cases of interest, such as (1) velocity gradients generating traveltime triplications and shadows, (2) wave-front bending, kinking and folding in a low-velocity waveguide, and (3) wavefield propagation perturbed by a high-velocity slab. The results show that many features of high- and intermediate-frequency waveforms are reliably predicted by Maslov’s technique, but also that it is far less reliable and even fails for low frequencies. The terms ‘high’ and ‘low’ are model-dependent, but we mean the range over which it is sensible to discuss signals associated with identifiable wave fronts and local (if complicated) eVects that potentially can be unravelled in interpretation. Of the high- and intermediate-frequency wave components, those wavefront anomalies due to wave-front bending, kinking, folding or rapid ray divergence can be accurately given by MI. True diVractions due to secondary wave-front sections are theoretically not included in Maslov theory (as they require true diVracted rays), but in practice they can often be satisfactorily predicted. This occurs roughly within a wavelength of the truncated geometrical wave front, where such diVractions are most important since their amplitudes may still be as large as half that on the geometrical wave front itself. Outside this region MI is inaccurate (although then the diVractions are usually small). Thus waveforms of high and intermediate frequencies are essentially controlled by classical wave-front geometry. Our results also show that the accuracy of MI can be improved by rotating the Maslov integration axis so that the nearest wave-front anomaly is adequately ‘sampled’. This rotation can be performed after ray tracing and it can serve to avoid pseudocaustics by using it in conjunction with the phase-partitioning approach. The eVort needed in phase partitioning has been reduced by using an interactive graphics technique. It is diYcult to formulate a general rule prescribing the limitations of MI accuracy because of model dependency. However, our experiences indicate that two space- and two timescales need to be considered. These are the pulse width in space, the length scale over which the instantaneous wave-front curvature changes, and the timescales of pulse width and significant features in the ray traveltime curve. It seems, from our examples, that when these scales are comparable, the Maslov method gives very acceptable results.

SUMMARY 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.

Summary 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. Cervený 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.

SUMMARY 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 X 2 matrices not 3 x 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.

SUMMARY 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.