The methodology of marching phase-space transformations has been previously considered for slowly varying media. The present work implements the approach for a medium with a planar wave velocity discontinuity. It is shown that propagation in the water column can be systematically separated from the scattering events at the discontinuity. The latter take place in a narrow interaction zone, out of which the previously invoked local homogeneity assumption can be used again. The width of the interaction zone is obtained by the investigation of the recently developed evolution equation of the local spectra. The associated phase-space propagators and the isolation of phenomena such as reflection, refraction, and head waves in phase space are demonstrated. Computations are performed and compared to reference solutions.