By introducing a splitting into up and down going waves, linear wave equations can be rewritten as coupled first-order systems in the up and down going components of the field. Exact local splittings yield uncoupled systems wherever the medium does not vary in the preferred direction, but approximate splittings can be more convenient. Given a slab of an inhomogeneous medium and a splitting, one can define an associated scattering matrix. Invariant imbedding techniques allow one to write a coupled system of differential equations for the operator entries of the scattering matrix, where the differentation is with respect to the location of one of the planes of the slab. One can then deduce the behavior of the reflection operator for small times, which provides a connection between the up and down going fields and the properties of the medium on the edge of the slab. Downward continuation inversion algorithms can then be derived. An example is given for the electromagnetic wave equation. Dissipative effects can be modelled, and are further studied in the companion paper, "The effects of dissipation in one-dimensional inverse scattering problems."