Mathematics of Modeling, Migration and Inversion with Gaussian Beams

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Gaussian beams are extensions of asymptotic ray theory to waves with complex traveltime. These waves have Gaussian decay orthogonal to a central ray. Solutions of wave problems are represented by integrals (sums) over a suite of Gaussian beams. This tends to produce representations that are smoother than those produced by symptotic ray theory, facilitating smoother modeling, migration and inversion output than what is produced by classical asymptotic ray theory. Asymptotic ray theory is reviewed in the hierarchy of Cartesian coordinates, ray-centered coordinates, ray-centered coordinates with complex traveltimes (Gaussian beams!). Green’s functions and plane-wave modeling are described in each case, with the last requiring integrals over suites of Gaussian beams. Examples of Kirchhoff migration/inversion using Gaussian beam representations of Green’s functions are presented.