We present a generalised framework for focusing functions by introducing the conceptual idea of sources that propagate energy likewise forwards and backwards in time. These functions can be used to extend the concept of homogeneous Green’s functions. Based on the resulting partial differential equation we develop integral representations for both closed and open boundaries that relate focusing and Green’s functions. The open boundary representation is similar to that obtained in the context of the recent Marchenko scheme, however, our new derivation imposes significantly fewer limitations on the involved wavefields. In particular we do not require an auxiliary truncated medium (or any fields relating to one), nor do we require up-/down-decomposition at depth. Thus, we ultimately present a Marchenko integral that allows for retrieving full wavefield Green’s functions, involving refracted, diving and evanescent waves.

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