The ill-posed elliptic wave propagation problems can be transformed into well-posed initial value problems of the reflection and transmission operators characterizing the material structure of the given model by the combination of wave field splitting and invariant imbedding methods. In general, the derived scattering operator equations are of first-order in range, nonlinear, nonlocal, and stiff and oscillatory with a subtle fixed and movable singularity structure. The phase space and path integral analysis reveals that construction and reconstruction algorithms depend crucially on a detailed symbol analysis of the scattering operators. Some information about the singularity structure of the scattering operator symbols is presented and analyzed in the transversely homogeneous limit.