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 subsequent equations for the reflection and transmission operators are of first-order (one-way) in range, nonlinear (Riccati-like), and, in general, nonlocal. The reflection and transmission operator equations provide the framework for constructing inverse algorithms based on, in principle, exact solution methods.