Direct and inverse wave propagation theories and the factorized Helmholtz equation

Abstract

The reduced scalar Helmholtz equation for a transversely inhomogeneous half-space supplemented with an outgoing radiation condition and an appropriate boundary condition on the initial-value plane defines a direct acoustic propagation model. This elliptic formulation admits a factorization and is subsequently equivalent to a first-order Weyl pseudo-differential equation which is recognized as an extended parabolic propagation model. Perturbation treatments of the appropriate Weyl composition equation result in a systematic development of approximate wave theories while exact inversions for several nontrivial profiles provide for an analysis of strong refractive and diffractive effects. The analysis, in a natural manner, provides for the formulation and exact solution of a multidimensional nonlinear inverse problem appropriate for ocean acoustic, seismic, and optical studies. Backscatter effects in both the direct and inverse problems can be included through weak-backscatter and spatial-dimensional (imbedding) perturbation methods. Moreover, the n-dimensional reduced scalar Helmholtz equation for the transversely inhomogeneous medium is naturally related to parabolic propagation models through (1) the above mentioned n-dimensional extended parabolic (Weyl pseudo-differential) equation and (2) an imbedding in an (n + 1)-dimensional parabolic (Schrödinger) equation. The first relationship provides the basis for the parabolic-based Hamiltonian phase space path integral representation of the half-space propagator. The second relationship provides the basis for the elliptic-based path integral representations associated with Feynman/Fradkin, Feynman/Garrod, and Feynman/Dewitt-Morette. The path integrals allow for a global perspective of the transition from elliptic to parabolic wave theory in addition to providing a unifying framework in the direct and inverse formulations for dynamical approximations, resolution of the square root operator, and the concepts of an underlying stochastic process and free motion on curved spaces, the wave equation and path integral analysis foreshadows computational algorithms, extension to (1) the vector formulation appropriate for elastic media and (2) the bilinear formulation appropriate for acoustic field coherence, and application to wave propagation in random media.