Modes to replace transitional asymptotic ray fields in a vertically inhomogeneous earth model

Summary. Asymptotic ray theory (ART) fails in transition regions near critically reflected, bottom glancing or caustic-forming rays in a vertically inhomogeneous layered earth. These deficiencies are repaired here by replacing the transitional ray fields with guided modes plus truncation remainders. Exact ray-mode equivalences and their high-frequency asymptotic approximations are formulated, and their validity and efficiency are verified by numerical comparisons for SH motion in a two-layer earth model comprised of an inhomogeneous sediment above an homogeneous semi-infinite bedrock.     

Real and complex spectra – a generalization of WKBJ seismograms

Real plane-waves constitute the building blocks for recently developed spectral techniques in synthetic seismology. While providing numerical convenience, real slowness-spectra model certain wave phenomena in a distributed 'unnatural' way, whereas complex spectra model these phenomena in a compact, more 'natural' way. The theory of complex spectra, called by us the 'Spectral Theory of Transients' (STT) and developed elsewhere, is summarized here and contrasted with the real-spectrum approach. Relying strongly on the theory of analytic functions, STT permits the transient responses to be classified and evaluated according to the singularities they introduce in the complex slowness plane. The method is illustrated for a number of 2-D SH -wave model propagation environments, including interface reflection, head waves, multiple encounters with caustics due to concave boundaries or ducting medium inhomogeneities, and diffraction by structures with edges.     

Target strength: Some recent theoretical developments

Some recent developments in the theory of diffraction of transient wavefields with high or moderate, but not excessively low, frequency content are reviewed here. Examples are cited to demonstrate that combinations of wavefronts and resonances, which organize the wave process according to progressing and oscillatory events, respectively, can furnish building blocks for interpretation of transient scattering data, and thereby for classification of target features. Conversely, these same spectral elements hold promise as a basis for physically sound parameterization of scattering data to reconstruct target characteristics. The discussion includes impenetrable and penetrable targets, interior-exterior coupling, real and complex spectra, and suggested directions for future research.     

Matrix Green's functions for array-type sources and receivers in multiwave layered media

Summary. A previous formulation (Lu, Felsen & Kamel) of source-excited wave propagation in a multiwave layer is here extended to multiple layers, each of which may propagate different multiple wave species, and to simultaneous excitation and detection at arbitrarily specified multiple levels. Field variables are arranged so as to reveal 'interesting'layers requiring access (for example, those containing a source and/or receiver) but to hide in collective form all other 'uninteresting'layers. An ordering of wave constituents into array vectors provides not only a physically appealing view of the wave phenomena pertaining to array-type source and receiver arrangements but may also furnish numerical advantages. The variety of alternative representations in Lu et al. can be brought to bear directly on the present formulation which is thereby endowed with substantial versatility, especially that embodied within the hybrid ray-mode format.     

Spectral aspects of the Gaussian beam method: reflection from a homogeneous half-space

Summary. Use of paraxially approximated Gaussian beams continues to be actively pursued for construction of synthetic seismograms in complicated environments. How to select the beams in the stack remains a source of difficulty which has primarily been addressed by semi-heuristic considerations. In this paper, the classical example of line-source field reflection from a homogeneous half-space that can sustain a head wave is examined from a plane-wave spectral point of view. The individual beam fields are modelled exactly by the complex source point technique, which emphasizes the complex spectral content of these wave objects. The quality of the paraxial approximation of a typical reflected (Gaussian) beam characterized by different parameters is examined from this perspective, and is compared with uniform and non-uniform asymptotics generated from the exact beam field spectral integral. With this information as background, the reflected field for a real line-source is synthesized by beam superposition. Except for the immediate vicinity of the critical reflection angle, the well-known failure of narrow paraxial beams, no matter how densely stacked, to reproduce the head wave effects is shown to be due to the inadequate spectral content of these beams and not to the failure of beam stacking per se. When the rigorous solutions are used for the narrow-waist beams, even relatively few suffice to yield agreement with the exact solution. This circumstance emphasizes the importance of fully understanding the spectral implications of various beam stacking schemes.     

Geometrical theory of diffraction, evanescent waves, complex rays and Gaussian beams

Summary. The Gaussian beam method has recently been introduced into synthetic seismology to overcome shortcomings of the ray method, especially in transition regions due to focusing or diffraction where ray theory fails. One proceeds by discretizing the initial data as a superposition of paraxial Gaussian beams, each of which is then traced through the seismic environment. Since Gaussian beam fields do not diverge in ray transition regions, they are 'uniformly regular' although the quality of this regularity depends on the beam parameters and on the 'numerical distance' which defines the extent of the transitional domain. However, when Gaussian beam patches are used to simulate non-Gaussian initial data, there arise ambiguities due to choice of patch size and location, beam width, etc., which are at the user's disposal. The effects of this arbitrariness have customarily been explored by trial and error numerical experiment but no quantitative recommendations have emerged as yet. As a step toward a priori predictive capability, it is proposed here to perform a systematic study on analytically tractable prototype models of how the parameters and location of a single beam affect the quality of the observed seismic field, especially in ray transition regions. The conversion of ordinary ray fields into beam fields in canonical configurations can be accomplished conveniently by displacing a real source point into a complex coordinate space. Thus, the desired beam solutions can be obtained directly from available ray, and even paraxial ray, fields. Complex ray theory and its implications are reviewed here, with an emphasis on improvements of beam tracking schemes employed at present.