Wave field splitting, invariant imbedding, and phase space methods reformulate the Helmholtz wave propagation problem in terms of an operator scattering matrix characteristic of the modeled environment. The equations for the reflection and transmission operators are first-order in range, nonlinear (Riccati-like), and, in general, nonlocal. The singularity structure of the corresponding Weyl operator symbols plays a crucial role in the development of both direct and inverse wave propagation algorithms.

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