Applications of Directional Wavefield Decomposition, Phase Space, and Path Integral Methods to Seismic Wave Propagation and Inversion

— Recently, de Hoop and coworkers developed an asymptotic, seismic inversion formula for application in complex environments supporting multi-pathed and multi-mode wave propagation (de Hoop et al., 1999; de Hoop and Brandsberg-Dahl, 2000; Stolk and de Hoop, 2000). This inversion is based on the Born/Kirchhoff approximation, and employs the global, uniform asymptotic extension of the geometrical method of “tracing rays” to account for caustic phenomena. While this approach has successfully inverted the multicomponent, ocean-bottom data from the Valhall field in Norway, accounting for severe focusing effects (de Hoop and Brandsberg-Dahl, 2000), it is not able to account properly for wave phenomena neglected in the “high-frequency” limit (i.e., diffraction effects) and strong scattering effects. To proceed further and incorporate wave effects in a nonlinear inversion scheme, the theory of directional wavefield decomposition and the construction of the generalized Bremmer coupling series are combined with the application of modern phase space and path (functional) integral methods to, ultimately, suggest an inversion algorithm which can be interpreted as a method of “tracing waves.” This paper is intended to provide the seismic community with an introduction to these approaches to direct and inverse wave propagation and scattering, intertwining some of the most recent new results with the basic outline of the theory, and culminating in an outline of the extended, asymptotic, seismic inversion algorithm. Modeling at the level of the fixed-frequency (elliptic), scalar Helmholtz equation, exact and uniform asymptotic constructions of the well-known, and fundamentally important, square-root Helmholtz operator (symbol) provide the most important results.