HR: 16:00h
AN: S22G-01    [PDF]
TI: Coherent-state analysis of the seismic head-wave problem: an overcomplete representation and its relationship to rays and beams
AU: * Thomson, C J
EM: thomson@geol.queensu.ca
AF: Queen's University, Geological Sciences \& Geological Engineering, Kingston, ON K7L 3N6 Canada
AB: The coherent-state transform (CST) is a Gaussian-windowed Fourier transform and compared to the usual plane-wave expansion of seismic wavefields it provides an overcomplete basis of partial waves. These overcomplete partial waves have associated rays and the relationship of these rays to those of a physical wavefront is explained here by appealing to a familiar analytic example. The exact CST of the standard Fourier plane-wave summation for a point-source primary wave can be interpreted as a sum of damped plane waves forming a focussed or beam-like wavefield. This primary-wave CST can also be approximated as a bundle of complex rays in a position-slowness space with higher dimension than that usually considered, a consequence of the overcompleteness. The higher-dimensional ray spreading controls the coherent-state (CS) amplitude. This complex-ray bundle in turn can be approximated as a paraxial Gaussian beam carried by a real ray associated with the `physical' wavefront. The exact CST of the standard plane-wave summation for a point-source reflection shows how the complex-ray and paraxial-beam approximations generalize to interfaces. Around a critical angle the standard plane-wave summation has an asymptotic form involving the Weber function and this function also necessarily arises in the complex-ray and paraxial-beam approximations for the reflection CST. The inverse CST giving the point-source wavefield is a sum of coherent states and those contributing rays or paraxial beams which are incident around the critical angle should be given a reflection coefficient involving the Weber function. CS beams which are incident away from the critical angle collect a peripheral branch-point diffraction. In general, both types of critical-ray/branch-point signal should be included if the integrated CS response is to correctly describe the head wave. An advantage of the CST is that it combines with ray theory for gradually-varying media to give a convenient solution to the caustic and pseudocaustic problems. The general idea of such overlapping partial-wave expansions, their flexibility and the smoothing they impart may have other applications in seismic analysis and processing.
DE: 7203 Body wave propagation
DE: 7260 Theory and modeling
SC: Seismology [S]
MN: 2003 Fall Meeting