Effects of aperture on Marchenko focussing functions and their radiation behaviour at depth.

A focussing function is a specially constructed field that focusses on to a purely downgoing pulse at a specified subsurface position upon injection into the medium. Such focussing functions are key ingredients in the Marchenko method and in its applications such as retrieving Green's functions, redatuming, imaging with multiples and synthesizing the response of virtual sources/receiver arrays at depth. In this study, we show how the focussing function and its corresponding focussed response at a specified subsurface position are heavily influenced by the aperture of the source/receiver array at the surface. We describe such effects by considering focussing functions in the context of time-domain imaging, offering explicit connections between time processing and Marchenko focussing. In particular, we show that the focussed response radiates in the direction perpendicular to the line drawn from the centre of the surface data array aperture to the focussed position in the time-imaging domain, that is, in time-migration coordinates. The corresponding direction in the Cartesian domain follows from the sum (superposition) of the time-domain direction and the directional change due to time-to-depth conversion. Therefore, the result from this study provides a better understanding of focussing functions and has implications in applications such as the construction of amplitude-preserving redatuming and imaging, where the directional dependence of the focussed response plays a key role in controlling amplitude distortions.