Effects of aperture on Marchenko focusing functions and their radiation direction

A focusing function is a specially constructed field that will come into focus at a specified subsurface position upon back propagation (injection) into the medium. The concept of focusing functions is a key ingredient in the Marchenko method and its applications such as retrieving Green's function, redatuming, imaging with multiples, and creating virtual sources/receivers. In this study, we show how the focusing function and its corresponding focused response at a specified subsurface position are heavily influenced by the data aperture at the surface. Such effects can be explained by considering focusing function in the context of time-domain imaging and its usual assumptions. In particular, we show that the focused response in the time-imaging domain radiates in the direction perpendicular to the line drawn from the center of the surface data aperture to the focused position. The corresponding direction in the Cartesian domain is simply then a combination of the time-domain direction and the directional change due to time-to-depth conversion. Therefore, the result from this study provides insights towards a better understanding of focusing function and may have meaningful implications in applications such as the construction of virtual subsurface source, where the directionality of the focused response is important.