The Gel'fand-Levitan equation, the Gopinath-Sondhi equation, and the Marchenko equation are developed for one-dimensional inverse scattering problems. Recently, a version of the Marchenko equation based on wavefield decomposition has been introduced for focusing waves in multi dimensions. However, wavefield decomposition is a limitation when waves propagate horizontally at the focusing level. Here, the Marchenko equation for focusing without wavefield decomposition is derived, and by iteratively solving the Marchenko equation, the Green's function for an arbitrary location in the medium is retrieved from the scattered waves recorded on a closed receiver array and an estimate of the direct-wave without wavefield decomposition.

Marchenko algorithms retrieve the Green’s function for arbitrary subsurface locations, and the retrieved Green’s function includes the primary and multiple reflected waves. The Marchenko algorithms require the estimate of the direct arrivals and the reflected waves; however, most previous Marchenko algorithms also require the up/down components of the Marchenko equation for the Green’s function retrieval. We use the Marmousi model to retrieve the Green’s function without using the up/-down components of the Marchenko equation and show that the retrieved Green’s function matches with the numerically modeled Green’s function. We also show that the refracted waves can be successfully produced independently from the acquisition geometry, i.e., singlesided or two-sided; however, the retrieval of refracted waves that arrive before the first primary waves is inconsistent with the requirement that the Green’s function vanishes before the direct wave. Even though we retrieve such refracted waves, they are caused by the injection of the direct wave into suciently detailed background velocity and density models instead of operations of the Marchenko algorithm on the recorded wavefields.