Fundamentals of Seismic Wave Propagation: Frontmatter

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

Waveguides formed between the earth's free-surface and sharp subsurface velocity discontinuities, such as the watertable, may have a significant impact on high-resolution seismic refraction and reflection data. Energy trapped in such waveguides may propagate over long distances with little loss due to geometrical spreading. The dispersive nature of the guided waves results in pronounced shingling of the first arrivals. This narrows the “optimum reflection window” between the first arrivals and the ground roll, and may lead to mispicking of first breaks and difficulties in identifying and processing shallow reflections.

Effects of near‐surface waveguides on shallow high‐resolution seismic refraction and reflection data

The mathematical theory which is typically used to model the intrinsic anelasticity of the earth is the linear theory of viscoelasticity. The effects of anelasticity on wave propagation, such as absorption and dispersion, are often described using one‐dimensional (1-D) plane waves of the form exp[i(ωt-kx)] with k complex and frequency‐dependent. These waves are solutions of the 1-D viscoelastic wave equation. The reflection and transmission of plane waves in a layered viscoelastic medium is, however, a 2-D or 3-D problem. The solutions to the 2-D or 3-D viscoelastic wave equation are the so‐called general plane waves, which are classified as homogeneous or inhomogeneous depending upon whether or not the planes of constant phase, i.e., wavefronts, coincide with the planes of constant amplitude (the 1-D plane waves mentioned above are strictly homogeneous).

On the reflection and transmission of viscoelastic waves- Some numerical results

Diffraction and anelasticity problems involving decaying, “evanescent” or “inhomogeneous” waves can be studied and modelled using the notion of “complex rays”. The wavefront or “eikonal” equation for such waves is in general complex and leads to rays in complex position-slowness space. Initial conditions must be specified in that domain: for example, even for a wave originating in a perfectly elastic region, the ray to a real receiver in a neighbouring anelastic region generally departs from a complex point on the initial-values surface. Complex ray theory is the formal extension of the usual Hamilton equations to complex domains. Liouville's phase-space-incompressibility theorem and Fermat's stationary-time principle are formally unchanged. However, an infinity of paths exists between two fixed points in complex space all of which give the same final slowness, travel time, amplitude, etc. This does not contradict the fact that for a given receiver position there is a unique point on the initial-values surface from which this infinite complex ray family emanates.

Complex Rays and Wave Packets for Decaying Signals in Inhomogeneous, Anisotropic and Anelastic Media

Exact synthetic seismograms are obtained for a simple layered elastic half‐space due to a buried point force and a point torque. Two models, similar to those encountered in seismic exploration of sedimentary basins, are examined in detail. The seismograms are complete to any specified time and make use of a Cagniard‐Pekeris method and a decomposition into generalized rays. The weathered layer is modeled as a thin low‐velocity layer over a half‐space. For a horizontal force in an arbitrary direction, the transverse component, in the near‐field, shows detectable first arrivals traveling with a compressional wave velocity. The radial and vertical components, at all distances, show a surface head wave (sP*) which is not generated when the source is compressive. A buried vertical force produces the same surface head wave prominently on the radial component. An example is given for a simple “Alberta” model as an aid to the interpretation of wide angle seismic reflections and head waves.

Exact seismograms for a point force using generalized ray theory

Attenuation and dispersion are included in synthetic seismograms obtained by a Cagniard‐Pekeris formulation for the problem of a point source in a layer over a half‐space. The solution is decompose into generalized rays, and the effects of attenuation and dispersion are incorporated in an ad hoc manner in the frequency domain. The effects of the viscoelastic interfaces are taken into account by corrections to the reflection coefficient for an elastic medium. The results are illustrated with synthetics for a model simulating a weathered layer over a halt‐space. Both the SH and P‐SV cases are treated.

Attenuation of seismograms obtained by the Cagniard‐Pekeris method

The displacement components for a horizontal stress discontinuity along a buried finite fault in an elastic homogeneous layer on top of an elastic half-space are given analytically in terms of generalized rays. For a particular case of a concentrated horizontal force pointing in an arbitrary direction, detailed time-dependent expressions are given. For a simple model of a “crustal” layer over a “mantle” half-space, the numerical seismograms in the near- and intermediate-field show some interesting features. These include a prominent group of compressional waves whose radial component is substantial at distances four times the crustal thickness. All the dominant shear arrivals ( s , SS , and sSS ) are important and show large variations of amplitude as the source depth and receiver distance are varied. Some of the prominent individual generalized rays are shown, and it is found that they can be grouped naturally into families based on the number of interactions with the boundaries. The subdivision into individual generalized rays is useful for analysis and for checks on the numerical stability of the synthetic seismograms. Since the solution is analytic and the numerical evaluation is complete up to any desired time, the results are useful in comparing other approximate methods for the computation of seismograms.

P. G. Kelamis

Papers

Exact seismograms for a point force using generalized ray theory

Attenuation of seismograms obtained by the Cagniard‐Pekeris method

Seismic waves from a horizontal stress discontinuity in a layered solid

Synthetic Seismograms With Attenuation For Buried Compressional And Shear Sources