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Many seismic imaging methods use wave field extrapolation operators toredatum sources and receivers from the surface into the subsurface. We discusswave field extrapolation operators that account for internal multiplereflections, in particular propagator matrices, transfer matrices and Marchenkofocusing functions.  A propagator matrix is a square matrix that `propagates' a wave-field vectorfrom one depth level to another. It accounts for primaries and multiples andholds for propagating and evanescent waves.  A Marchenko focusing function is a wave field that focuses at a designatedpoint in space at zero time. Marchenko focusing functions are useful forretrieving the wave field inside a heterogeneous medium from the reflectionresponse at its surface. By expressing these focusing functions in terms of thepropagator matrix, the usual approximations (such as ignoring evanescent waves)are avoided.  While a propagator matrix acts on the full wave-field vector, a transfermatrix (according to the definition employed in this paper)`transfers' adecomposed wave-field vector (containing downgoing and upgoing waves) from onedepth level to another. It can be expressed in terms of decomposed Marchenkofocusing functions.  We present propagator matrices, transfer matrices and Marchenko focusingfunctions in a consistent way and discuss their mutual relations. In the maintext we consider the acoustic situation and in the appendices we discuss otherwave phenomena. Understanding these mutual connections may lead to newdevelopments of Marchenko theory and its applications in wave field focusing,Green's function retrieval and imaging.

Propagator and transfer matrices, Marchenko focusing functions and their mutual relations