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Seismic methods are used in a wide variety of contexts to investigate subsurface Earth structures, and to explore and monitor resources and waste-storage reservoirs in the upper ~100km of the Earth's subsurface. Reverse-time migration (RTM) is one widely used seismic method which constructs high-frequency images of subsurface structures. Unfortunately, RTM has certain disadvantages shared with other conventional single-scattering-based methods, such as not being able to correctly migrate multiply-scattered arrivals. In principle, the recently-developed Marchenko methods can be used to migrate all orders of multiples correctly. In practice however, using Marchenko methods are costlier to compute than RTM—for a single imaging location, the cost of performing the Marchenko method is several times that of standard RTM, and performing RTM itself requires dedicated use of some of the largest computers in the world for individual datasets. A different imaging strategy is therefore required. We propose a new set of imaging methods which use so-called focusing functions to obtain images with few artifacts from multiply-scattered waves, while greatly reducing the number of points across the image at which the Marchenko method need be applied. Focusing functions are outputs of the Marchenko scheme: they are solutions of wave equations that focus in time and space at particular surface or subsurface locations. However, they are mathematical rather than physical entities, being defined only in reference media that equal to the true Earth above their focusing depths but are homogeneous below. Here, we use these focusing functions as virtual source/receiver surface seismic surveys, the upgoing focusing function being the virtual received wavefield that is created when the downgoing focusing function acts as a spatially distributed source. These source/receiver wavefields are used in three imaging schemes: one allows specific individual reflectors to be selected and imaged. The other two schemes provide either targeted or complete images with distinct advantages over current reverse-time migration methods, such as fewer artifacts and artifacts that occur in different locations. The latter property allows the recently published "combined imaging" method to remove almost all artifacts. We show several examples to demonstrate the methods: acoustic 1D and 2D synthetic examples, and a 2D line from an ocean bottom cable (OBC) field dataset. We discuss an extension to elastic media, which is illustrated by a 1.5D elastic synthetic example.

S U M M A R Y Seismic interferometry comprises a suite of methods to redatum recorded wavefields to those that would have been recorded if different sources (so-called virtual sources) had been activated. Seismic interferometry by cross-correlation has been formulated using either two-way (for full wavefields) or one-way (for directionally decomposed wavefields) representation theorems. To obtain improved Green’s function estimates, the cross-correlation result can be deconvolved by a quantity that identifies the smearing of the virtual source in space and time, the so-called point-spread function. This type of interferometry, known as interferometry by multidimensional deconvolution (MDD), has so far been applied only to one-way directionally decomposed fields, requiring accurate wavefield decomposition from dual (e.g. pressure and velocity) recordings. Here we propose a form of interferometry by multidimensional deconvolution that uses full wavefields with two-way representations, and simultaneously invert for pressure and (normal) velocity Green’s functions, rather than only velocity responses as for its one-way counterpart. Tests on synthetic data show that two-way MDD improves on results of interferometry by cross-correlation, and generally produces estimates of similar quality to those obtained by one-way MDD, suggesting that the preliminary decomposition into upand downgoing components of the pressure field is not required if pressure and velocity data are jointly used in the deconvolution. We also show that constraints on the directionality of the Green’s functions sought can be added directly into the MDD inversion process to further improve two-way multidimensional deconvolution. Finally, as a by-product of having pressure and particle velocity measurements, we adapt oneand two-way representation theorems to convert any particle velocity receiver into its corresponding virtual dipole/gradient source by means of MDD. Thus data recorded from standard monopolar (e.g. marine) pressure sources can be converted into data from dipolar (derivative) sources at no extra acquisition cost.