Review paper: Virtual sources and their responses, Part II: data-driven single-sided focusing

Abstract

In Part I of this paper, we defined a focusing wave field as the time reversal of an observed point-source response. We showed that emitting a time-reversed field from a closed boundary yields a focal spot that acts as an isotropic virtual source. However, when emitting the field from an open boundary, the virtual source is highly directional and significant artefacts occur related to multiple scattering. The aim of this paper is to discuss a focusing wave field, which, when emitted into the medium from an open boundary, yields an isotropic virtual source and does not give rise to artefacts. We start the discussion from a horizontally layered medium and introduce the single-sided focusing wave field in an intuitive way as an inverse filter. Next, we discuss singlesided focusing in two-dimensional and three-dimensional inhomogeneous media and support the discussion with mathematical derivations. The focusing functions needed for single-sided focusing can be retrieved from the single-sided reflection response and an estimate of the direct arrivals between the focal point and the accessible boundary. The focal spot, obtained with this single-sided data-driven focusing method, acts as an isotropic virtual source, similar to that obtained by emitting a time-reversed point-source response from a closed boundary.