Preface 1. Introduction 2. The elastodynamics and its simple solutions 3. Seismic rays and travel times 4. Dynamic ray tracing. Paraxial ray methods 5. Ray amplitudes 6. Ray synthetic seismograms Appendix. Fourier transform, Hilbert transform and analytical signals References Index.

Seismic Ray Theory

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/98RG02075

Seismic anisotropy and mantle deformation: What have we learned from shear wave splitting?

Volume II J. S. R. Chisholm and Rosa M. Morris Amsterdam: North-Holland. 1964 Pp. xviii + 717. Price 72s. This book is a valuable addition to an ever-growing collection of mathematical texts written expressly for the physicist. The authors suggest that the book might also be useful to undergraduates in chemistry and engineering and research workers in other fields, but the content is clearly chosen to meet the needs of modern undergraduate physics.

http://iopscience.iop.org/article/10.1088/0031-9112/16/8/015/pdf

Mathematical Methods in Physics

Fundamentals of Seismic Wave Propagation: Frontmatter

Mathematical Methods of Physics.

Summary

We point out that while the equations of seismic ray geometrical spreading given recently by Norris do differ as stated from equations given by Cerveny, 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.

/pdf/a-comment-on-the-form-of-the-geometrical-spreading-equations-3ir1evgl2c.pdf

A comment on the form of the geometrical spreading equations, with some numerical examples of seismic ray tracing in inhomogeneous, anisotropic media

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.

Maslov Ray Summation, Pseudo-Caustics, Lagrangian Equivalence and Transient Seismic Waveforms

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’ (eg 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

/pdf/ray-theory-green-s-function-reciprocity-and-ray-centred-39bwqssccc.pdf

Ray-theory Green's function reciprocity and ray-centred coordinates in anisotropic media

Summary

A derivation is given for the factorization of the generally anisotropic, elastic wave equation into two operators which are first order with respect to a preferred direction of propagation. One factor gives the ‘one-way’ equation for forward-going body waves. A 3 × 3 matrix notation is used which emphasizes the role of the Christoffel equation (or what might be called its ‘preferred coordinate projection’) familiar from ray theory. To recognize this it helps to obtain first an exact factorization of the wave equation for homogeneous anisotropic media. When inhomogeneities exist it is necessary to use a Fourier representation for differential operators with respect to the transverse coordinates, but still the Christoffel equation is the key. The so-called ‘square root operator’ required in the factorization becomes more correctly the ‘root of a matrix operator  ratic equation’ intimately connected to the algebraic Christoffel equation.



The factorization does not assume narrow-angle propagation or involve an a priori reference phase. It includes the P and two S waves and the forward coupling between them. It remains valid at slowness-surface singularities such as conical points (also known as acoustic axes—a most stringent test) and for wave fronts which fold. In the frequency domain this wide-angle one-way equation involves Fourier synthesis with respect to transverse position/slowness and its time-domain analogue involves2-D slant stacking (Radon transformation), multiplication by a 3 × 3 ‘propagator’ matrix and an inverse 2-D slant stack.



The one-way equation can be solved ‘analytically’ by using ‘path integrals’. These in turn may be reduced by stationary-phase arguments to standard ray theory, Maslov and Kirchhoff representations. In this sense, the ‘gap’ between ray theory and the numerical solution of the full wave equation is bridged or filled in, at least for forward propagation.



Alternatively, the one-way wave equation can be solved by numerical ‘forward-stepping’ algorithms and here again a number of choices are available. Narrow-angle Taylor or rational approximations to the matrix propagator in the Fourier/slant-stack representation lead to the anisotropic analogues of the classic 15° and 45° partial differential equations. The integral wide-angle form is also reminiscent of the ‘split-step’and ‘elastic phase screen’ approaches of Tappert, Wu and others, suggesting other implementations.

The ‘gap’ between seismic ray theory and ‘full’ wavefield extrapolation

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.

/pdf/seismic-modelling-of-subduction-zones-with-inhomogeneity-and-nnlgs8pqr0.pdf

C. J. Thomson

Papers

A comment on the form of the geometrical spreading equations, with some numerical examples of seismic ray tracing in inhomogeneous, anisotropic media

Maslov Ray Summation, Pseudo-Caustics, Lagrangian Equivalence and Transient Seismic Waveforms

Ray-theory Green's function reciprocity and ray-centred coordinates in anisotropic media

The ‘gap’ between seismic ray theory and ‘full’ wavefield extrapolation

Seismic Modelling of Subduction Zones With Inhomogeneity and Anisotropy-I. Teleseismic P-Wavefront Tracking.