Derivation and application of extended parabolic wave theories. I. 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 pseudodifferential 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 which extend the narrow-angle, weak-inhomogeneity, and weak-gradient ordinary parabolic (Schrödinger) approximation. The analysis further provides for the formulation and exact solution of a multidimensional nonlinear inverse problem appropriate for ocean acoustic and seismic studies. The wave theories foreshadow computational algorithms, the inclusion of range-dependent effects, and the extension to (1) the vector formulation appropriate for elastic media and (2) the bilinear formulation appropriate for acoustic field coherence.