By solving the Marchenko equations, the Green’s function can be retrieved between a virtual receiver in the subsurface to points at the surface (no physical receiver is required at the virtual location). We extend the idea of these equations to retrieve the Green’s function between any two points in the subsurface; i.e, between a virtual source and a virtual receiver (no physical source or physical receiver is required at either of these locations). This Green’s function is called the virtual Green’s function and includes all the primaries, internal and free-surface multiples. Similar to the Marchenko Green’s function, we require the reflection response at the surface (single-sided illumination) and an estimate of the first arrival travel time from the virtual location to the surface.@en

Measuring attenuation on the basis of interferometric, receiver-receiver surface waves is a non-trivial task: the amplitude, more than the phase, of ensemble-averaged cross-correlations is strongly affected by non-uniformities in the ambient wavefield. In addition, ambient noise data are typically pre-processed in ways that affect the amplitude itself. Some authors have recently attempted to measure attenuation in receiver-receiver cross-correlations obtained after the usual pre-processing of seismic ambient-noise records, including, most notably, spectral whitening. Spectral whitening replaces the cross-spectrum with a unit amplitude spectrum. It is generally assumed that cross-terms have cancelled each other prior to spectral whitening. Cross-terms are peaks in the cross-correlation due to simultaneously acting noise sources, that is, spurious traveltime delays due to constructive interference of signal coming from different sources. Cancellation of these cross-terms is a requirement for the successful retrieval of interferometric receiver-receiver signal and results from ensemble averaging. In practice, ensemble averaging is replaced by integrating over sufficiently long time or averaging over several cross-correlation windows. Contrary to the general assumption, we show in this study that cross-terms are not required to cancel each other prior to spectral whitening, but may also cancel each other after the whitening procedure. Specifically, we derive an analytic approximation for the amplitude difference associated with the reversed order of cancellation and normalization. Our approximation shows that an amplitude decrease results from the reversed order. This decrease is predominantly non-linear at small receiver-receiver distances: at distances smaller than approximately two wavelengths, whitening prior to ensemble averaging causes a significantly stronger decay of the cross-spectrum.@en

Imaging an unknown object in a medium that is known, such as a medium with constant velocity, is not difficult because one knows exactly where the waves are when they interact with the object. It is much more challenging to image an object in an unknown medium, because in that case one may know the waves that one sends into the medium, but one may does not know the waves that illuminate the object because the waves are distorted during their propagation to the object and back. Yet in many applications the medium is strongly scattering and the wavefield is strongly distorted as it propagates to the object. This is like imaging through frosted glass. How can one create an image in such media? And related to this, how can one focus a wavefield through a complicated medium that one does not know? Inverse scattering methods, as developed in quantum mechanics[1, 2], make it possible to estimate the model or object at a prescribed location without knowing the medium between that location and the point where reflected waves are recorded. These inverse scattering methods are known as the Marchenko equation or Gel’Fand-Levitan equation. Recently, these inverse scattering methods have been generalized to applications in seismology[3, 4, 5, 6] where one seeks to image a target, such as a reservoir, under a complicated overburden, such as a salt body. The main issue we will address is how it is possible that one can image the object at one location without knowing the medium between the observation point and the reconstruction point. The reason why inverse scattering make it possible to do this is that these methods involve an integral equation[7], and the function that one solves for is akin to the Green’s function for the unknown medium. The function obtained by solving the Marchenko equation is, in fact, the incident wavefield that will focus the waves at a specified target location. In order to solve this integral equation one only needs to know a smooth estimate of he velocity model and the reflected waves recorded at the acquisition surface, but the details of the complexity of the medium need not be known. That means there exists a recipe to determine, given the reflected waves, the incident wavefield that focuses at a specified target point. Such focusing is exactly what is needed to determine the image at the target point. There are many applications in geophysics where one seeks to create an image in strongly scattering media. These include hydrocarbon reservoirs under a complicated overburden, the interior of volcanoes, possibly the core mantle boundary, and crustal structure from high-frequency seismic waves. @en

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. @en

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. @en

Imagine placing a receiver at any location in the earth and recording the response at that location to sources on the surface. In such a world, we could place receivers around our reservoir to better image the reservoir and understand its properties. Realistically, this is not a feasible approach for understanding the subsurface. We have developed an alternative and realizable approach to obtain the response of a buried virtual receiver for sources at the surface. Our method is capable of retrieving the Green’s function for a virtual point in the subsurface to the acquisition surface. In our case, a physical receiver is not required at the subsurface point; instead, we require the reflection measurements for sources and receivers at the surface of the earth and a macromodel of the velocity (no small-scale details of the model are necessary). We can interpret the retrieved Green’s function as the response to sources at the surface for a virtual receiver in the subsurface. We obtain this Green’s function by solving the Marchenko equation, an integral equation pertinent to inverse scattering problems. Our derivation of the Marchenko equation for the Green’s function retrieval takes into account the free-surface reflections present in the reflection response (previous work considered a response without free-surface multiples). We decompose the Marchenko equation into up- and downgoing fields and solve for these fields iteratively. The retrieved Green’s function not only includes primaries and internal multiples as do previous methods, but it also includes freesurface multiples. We use these up- and downgoing fields to obtain a 2D image of our area of interest, in this case, below a synclinal structure. @en

Virtual sources can be created in several ways. In seismic interferometry, a virtual source is created by crosscorrelating responses at different receivers, which are illuminated from all directions. Seismic interferometry can be mathematically described by the homogeneous Green's function representation, which is a closed boundary integral. Virtual sources can also be created with the Marchenko method. For the Marchenko method it is sufficient that the position of the virtual source is illuminated from one side. We derive a single-sided homogeneous Green's function representation, which is an open boundary integral along reflection measurements at the surface. Applying this representation, we obtain virtual sources and virtual receivers in the subsurface from real sources and receivers at the surface (note that in our earlier work on the Marchenko method the response to the virtual source was only obtained for receivers at the surface). The retrieved virtual data show the entire evolution of the response to a virtual source in the subsurface, including primary and multiple scattering at unknown interfaces. @en

Virtual sources can be created in several ways. In seismic interferometry, a virtual source is created by crosscorrelating responses at different receivers, which are illuminated from all directions. Seismic interferometry can be mathematically described by the homogeneous Green's function representation, which is a closed boundary integral. Virtual sources can also be created with the Marchenko method. For the Marchenko method it is sufficient that the position of the virtual source is illuminated from one side. We derive a single-sided homogeneous Green's function representation, which is an open boundary integral along reflection measurements at the surface. Applying this representation, we obtain virtual sources and virtual receivers in the subsurface from real sources and receivers at the surface (note that in our earlier work on the Marchenko method the response to the virtual source was only obtained for receivers at the surface). The retrieved virtual data show the entire evolution of the response to a virtual source in the subsurface, including primary and multiple scattering at unknown interfaces. @en

Virtual sources can be created in several ways. In seismic interferometry, a virtual source is created by crosscorrelating responses at different receivers, which are illuminated from all directions. Seismic interferometry can be mathematically described by the homogeneous Green's function representation, which is a closed boundary integral. Virtual sources can also be created with the Marchenko method. For the Marchenko method it is sufficient that the position of the virtual source is illuminated from one side. We derive a single-sided homogeneous Green's function representation, which is an open boundary integral along reflection measurements at the surface. Applying this representation, we obtain virtual sources and virtual receivers in the subsurface from real sources and receivers at the surface (note that in our earlier work on the Marchenko method the response to the virtual source was only obtained for receivers at the surface). The retrieved virtual data show the entire evolution of the response to a virtual source in the subsurface, including primary and multiple scattering at unknown interfaces. @en

Virtual sources can be created in several ways. In seismic interferometry, a virtual source is created by crosscorrelating responses at different receivers, which are illuminated from all directions. Seismic interferometry can be mathematically described by the homogeneous Green's function representation, which is a closed boundary integral. Virtual sources can also be created with the Marchenko method. For the Marchenko method it is sufficient that the position of the virtual source is illuminated from one side. We derive a single-sided homogeneous Green's function representation, which is an open boundary integral along reflection measurements at the surface. Applying this representation, we obtain virtual sources and virtual receivers in the subsurface from real sources and receivers at the surface (note that in our earlier work on the Marchenko method the response to the virtual source was only obtained for receivers at the surface). The retrieved virtual data show the entire evolution of the response to a virtual source in the subsurface, including primary and multiple scattering at unknown interfaces. @en

Virtual sources can be created in several ways. In seismic interferometry, a virtual source is created by crosscorrelating responses at different receivers, which are illuminated from all directions. Seismic interferometry can be mathematically described by the homogeneous Green's function representation, which is a closed boundary integral. Virtual sources can also be created with the Marchenko method. For the Marchenko method it is sufficient that the position of the virtual source is illuminated from one side. We derive a single-sided homogeneous Green's function representation, which is an open boundary integral along reflection measurements at the surface. Applying this representation, we obtain virtual sources and virtual receivers in the subsurface from real sources and receivers at the surface (note that in our earlier work on the Marchenko method the response to the virtual source was only obtained for receivers at the surface). The retrieved virtual data show the entire evolution of the response to a virtual source in the subsurface, including primary and multiple scattering at unknown interfaces. @en

Virtual sources can be created in several ways. In seismic interferometry, a virtual source is created by crosscorrelating responses at different receivers, which are illuminated from all directions. Seismic interferometry can be mathematically described by the homogeneous Green's function representation, which is a closed boundary integral. Virtual sources can also be created with the Marchenko method. For the Marchenko method it is sufficient that the position of the virtual source is illuminated from one side. We derive a single-sided homogeneous Green's function representation, which is an open boundary integral along reflection measurements at the surface. Applying this representation, we obtain virtual sources and virtual receivers in the subsurface from real sources and receivers at the surface (note that in our earlier work on the Marchenko method the response to the virtual source was only obtained for receivers at the surface). The retrieved virtual data show the entire evolution of the response to a virtual source in the subsurface, including primary and multiple scattering at unknown interfaces. @en

Recent work on autofocusing with the Marchenko equation has shown how the Green's function for a virtual source in the subsurface can be obtained from reflection data. The response to the virtual source is the Green's function from the location of the virtual source to the surface. The Green's function is retrieved using only the reflection response of the medium and an estimate of the first arrival at the surface from the virtual source. Current techniques, however, only include primaries and internal multiples. Therefore, all surface-related multiples must be removed from the reflection response prior to the Green's function retrieval. Here, we present a new scheme that includes primaries, internal multiples, and free-surface multiples. In other words, we retrieve the Green's function in the presence of the free surface. The information needed for the retrieval are the reflection response at the acquisition surface and an estimate of the first arrival at the surface from the virtual source. The reflection response, in this case, includes the free-surface multiples; this makes it possible to include these multiples in the imaging operator and it obviates the need for surface-related multiple elimination.@en