Factorization, path integral representations, and the construction of direct and inverse wave propagation theories

Abstract

The development of both direct and inverse wave propagation theories at the level of the reduced scalar Helmholtz equation is addressed. The principal constructive tool is the factorization analysis of the Helmholtz equation which derives from the physically suggestive picture of decoupled forward and backward wave propagation appropriate for transversely inhomogeneous environments. The factorization analysis, most significantly, provides for a more “microscopic” understanding of the Helmholtz wave propagation process. In so doing, the analysis underscores the intimate connection between the direct and inverse theories, provides an inherently multidimensional framework, and further serves as a focal point for a wide range of physical (the concept of an underlying stochastic process, the notion of strong and weak-coupling regimes, free motion on curved spaces) and mathematical (pseudo-differential operator theory, path integral methods, imbedding techniques) perspectives. The operator and path integral analysis provides for computational algorithms while foreshadowing extensions to more complex environments and other physical theories.