Bananas, Doughnuts and Seismic Traveltimes

Most of what we know about the 3-D seismic heterogeneity of the mantle is based upon ray-theoretical traveltime tomography. In this infinite-frequency approximation, a measured traveltime anomaly depends only upon the wavespeed along an infinitesimally thin geometrical ray between a seismic source and a seismographic station. In this lecture I shall describe a new formulation of the seismic traveltime inverse problem which accounts for the ability of a finite-frequency wave to ``feel'' 3-D structure off of the source-receiver ray. Finite-frequency diffraction effects associated with this off-ray sensitivity act to ``heal'' the corrugations that develop in a wavefront propagating through a heterogeneous medium. Ray-theoretical tomography is based upon the premise that a seismic wave ``remembers'' all of the traveltime advances or delays that it accrues along its path, whereas actual finite-frequency waves ``forget''. I shall describe a number of recent analytical and numerical investigations, which have led to an improved theoretical understanding of this phenomenon.