  We consider wave propagation problems in which there is a preferred direction
of propagation. To account for propagation in preferred directions, the wave
equation is decomposed into a set of coupled equations for waves that propagate
in opposite directions along the preferred axis. This decomposition is not
unique. We discuss flux-normalised and field-normalised decomposition in a
systematic way, analyse the symmetry properties of the decomposition operators
and use these symmetry properties to derive reciprocity theorems for the
decomposed wave fields, for both types of normalisation. Based on the
field-normalised reciprocity theorems, we derive representation theorems for
decomposed wave fields. In particular we derive double- and single-sided
Kirchhoff-Helmholtz integrals for forward and backward propagation of
decomposed wave fields. The single-sided Kirchhoff-Helmholtz integrals for
backward propagation of field-normalised decomposed wave fields find
applications in reflection imaging, accounting for multiple scattering.
