Derivation and application of extended parabolic wave theories. I. The factorized Helmholtz equation

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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.