Publisher Summary This chapter focuses on elastic wave propagation in stratified media. The development of the theory of elastic wave propagation in stratified media has been strongly influenced by the problems of seismic wave propagation and the nature of the seismograms recorded from earthquakes. For purely analytic developments of elastic wave propagation, the level of manageable algebraic complexity is reached in a model with one or two uniform layers overlying a uniform half space. This chapter shows how the excitation of elastic waves, within a horizontally stratified structure, can be conveniently developed in terms of reflection and transmission matrices. This procedure has allowed the construction of the full response of the medium or approximations with desired properties so that theoretical seismograms may be calculated for realistic distributions of elastic parameters. Although this development has been for isotropic media, nearly all the results apply directly to the case of full anisotropy if 3 × 3 reflection and transmission matrices allowing coupling between all wave types are employed. This development of the wavefield for both source and receiver within the stratification may be used for other classes of wave propagation.

Seismic Wave Propagation in Stratified Media

Summary. The computation of theoretical seismograms for models in which the elastic parameters and density vary only with depth (in a plane, cylindrical or spherical geometry) reduces to the solution of an ordinary differential equation plus the evaluation of inverse transformations. In principle, the problem is straightforward. In practice, many techniques and approximations can be used at each stage and many combinations and variants are possible. In this paper, we discuss a new method of evaluating the inverse transforms. Any method can be used to solve the differential equation and we only discuss a few analytic approximations to illustrate the new method. The inverse transformations are a frequency and wavenumber integral. Essentially four techniques can be used to evaluate these depending on the order of integration and whether the wavenumber integral is distorted from the real axis. Three of these have been widely used, but the technique of evaluating the frequency integral first and keeping the wavenumber real is new. In this paper, we discuss some of the advantages of this combination.

A new method for computing synthetic seismograms

[1] A general theory for transitions between sub-Rayleigh and intersonic rupture speeds is developed for faults governed by slip-weakening friction. The transition occurs when stresses moving at intersonic speeds ahead of expanding or accelerating sub-Rayleigh ruptures exceed the peak strength of the fault, initiating slip within a daughter crack. Upon reaching a critical nucleation length, the daughter crack becomes dynamically unstable, expanding into a self-sustaining intersonic rupture. This mechanism holds in both two and three dimensions. On faults with uniform properties, the seismic S ratio [S = (τp − τ0)/(τ0 − τr)], a measure of the initial loading stress, τ0, relative to the peak and residual strengths, τp and τr, respectively, must be smaller than some critical value for the transition to occur. The maximum S value for unbounded faults in three dimensions is 1.19, smaller than the value of 1.77 that Andrews (1985) has shown to govern the transition in two dimensions. The supershear transition length (i.e., how far the rupture propagates before reaching intersonic speeds) is proportional to a length scale arising from the friction law governing the nucleation and stability of the daughter crack. A sufficiently narrow fault width suppresses the transition; the critical width is approximately 0.8 times the transition length on an unbounded fault. The transition length is highly sensitive to the form of the slip-weakening law even when the associated fracture energies are identical. Heterogeneous propagation, in the form of abrupt accelerations or increases in stress-release rate, induces stress-wave radiation that can trigger transient bursts of intersonic propagation.

/pdf/conditions-governing-the-occurrence-of-supershear-ruptures-5fubyiitiq.pdf

Conditions governing the occurrence of supershear ruptures under slip-weakening friction

Impact-echo :: a method for flaw detection in concrete using transient stress waves

Synthetic seismograms, computed for realistic, horizontally stratified media, are now routinely used as an aid to seismic interpretation. This paper reviews the theoretical background to these methods and presents comparisons of the two popular algorithms, reflectivity and WKBJ seismograms, for a variety of earth models. The transformed wave equations are developed from the equations for a spherical, gravitating medium in a symmetric form suitable for body wave calculations. Four methods of solving these equations in general, inhomogeneous layers are described: the WKBJ and Langer asymptotic expansions and the WKBJ and Langer iterative solutions. Together with the earth-flattening transformation and the ray expansion, transformed solutions for body waves can then be obtained for realistic layered media. Four methods of inverting the frequency and wave number transformations are also described: the real and complex spectral and slowness methods. Although realistic seismic models are normally sufficiently complicated that numerical calculations are essential, before proceeding with numerical comparisons we have included a review of the canonical signals included in body wave seismograms. These analytic results for direct rays, partial and total reflections, turning rays on forward and reversed branches, head waves, interface waves, Airy caustics, and Fresnel and interface shadows are useful to anticipate and understand numerical problems and results. Finally, a comparison of Green's functions for crustal, mantle, and whole earth models, calculated using the WKBJ and reflectivity algorithms, is included.

The computation of body wave synthetic seismograms in laterally homogeneous media

The Generalized Ray Theory and Transient Responses of Layered Elastic Solids

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The Generalized Ray Theory and Transient Responses of Layered Elastic Solids