Source-receiver Marchenko redatuming: Obtaining virtual receivers and virtual sources in the subsurface

By solving the Marchenko equations, one can retrieve the Green’s function (Marchenko Green’s function) between a virtual receiver in the subsurface and points at the surface (no physical receiver is required at the virtual location). We extend the idea behind 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 it includes all primary, internal, and free-surface multiples. Similar to the Marchenko Green’s function, this virtual Green’s function requires the reflection response at the surface (single-sided illumination) and an estimate of the first-arrival traveltime from the virtual locations to the surface. These Green’s functions can be used to image the interfaces from above and below. INTRODUCTION In this paper, we retrieve the Green’s function between two points in the subsurface of the earth. We call these two points a virtual source and a virtual receiver pair. To retrieve the Green’s function at a virtual receiver for a virtual source, we require neither a physical source nor a physical receiver at the virtual source and receiver location. The requirements for the retrieval of this Green’s function are the reflection response for colocated physical sources and physical receivers at the surface (single-sided illumination) and a smooth version of the velocity model (no small-scale details of the model are necessary). For brevity, we define this Green’s function, i.e., the response of a virtual source recorded by a virtual receiver, as the virtual Green’s function. Similar ideas of retrieving Green’s function between two points have been proposed, notably, in seismic interferometry (Wapenaar, 2004; Bakulin and Calvert, 2006; Curtis et al., 2006, 2009; van Manen et al., 2006; Snieder et al., 2007; Curtis and Halliday, 2010) and in the Marchenko method (Broggini and Snieder, 2012; Broggini et al., 2012; Wapenaar et al., 2013, 2014; Slob et al., 2014; Singh et al., 2015, 2016). However, these methods (interferometry and the Marchenko method) have more restrictions in the sourcereceiver geometry, as discussed later, for the accurate retrieval of Green’s function than our proposed method. In seismic interferometry, we create virtual sources at locations where there are physical receivers. We also require a closed surface of sources to adequately retrieve the Green’s function. Unlike interferometry, a physical receiver or physical source is not needed by our method to create either a virtual source or a virtual receiver, and we only require single-sided illumination (a closed surface of sources is not needed). The Green’s function retrieved by the Marchenko equations is the response to a virtual source in the subsurface recorded by physical receivers at the surface (Broggini and Snieder, 2012; Broggini et al., 2012; Wapenaar et al., 2013, 2014; Slob et al., 2014; Singh et al., 2015, 2016). The Marchenko retrieved Green’s function requires neither a physical source nor a physical receiver at the virtual source location in the subsurface. Our algorithm retrieves the Green’s function (upgoing and downgoing at the receiver) for virtual sources and virtual receivers. The Marchenko-retrieved Green’s functions are limited to virtual sources in the subsurface recorded at the surface, but our algorithm presented in this paper is not restricted to recording on the surface for each virtual source. In our method, the response of the virtual source can be retrieved for a virtual reciever anywhere in the subsurface. Wapenaar et al. (2016) propose similar work to ours; however, our derivation of the Green’s function between two arbitrary points in the subsurface is used for imaging the subsurface with the upand downgoing virtual Green’s functions. In this paper, we discuss the theory of retrieving the virtual Green’s function. Our numerical examples are split into three sections: (1) a Manuscript received by the Editor 11 May 2016; revised manuscript received 6 November 2016; published online 23 March 2017. Center for Wave Phenomena, Colorado School of Mines, Department of Geophysics, Golden, Colorado, USA. E-mail: [email protected]; [email protected]. © 2017 Society of Exploration Geophysicists. All rights reserved. Q13 GEOPHYSICS, VOL. 82, NO. 3 (MAY-JUNE 2017); P. Q13–Q21, 17 FIGS., 1 TABLE. 10.1190/GEO2016-0074.1 D ow nl oa de d 06 /1 2/ 17 to 1 38 .6 7. 23 9. 17 8. R ed is tr ib ut io n su bj ec t t o SE G li ce ns e or c op yr ig ht ; s ee T er m s of U se a t h ttp :// lib ra ry .s eg .o rg / verification of our algorithm to demonstrate that we retrieve the upand downgoing virtual Green’s functions (using a 1D example for simplicity), (2) a complicated 1D example illustrating our algorithm accurately retrieves the Green’s function with and without the freesurface multiples, and (3) a 2D numerical example of the virtual Green’s function constructed in such a way that we create a wavefield with all the reflections and direct waves from a virtual source. This last numerical example has the discontinuities in the density and the velocity at different locations. We then demonstrate how to apply these retrieved virtual Green’s functions for imaging. THEORY To retrieve the Green’s function from a virtual receiver in the subsurface for sources on the surface, one solves the Marchenko equations. The retrieval only requires the reflection response at the surface and an estimate of the first-arrival traveltime from the virtual receiver to the surface. The retrieved Green’s function can either include free-surface multiples (Singh et al., 2015, 2016) or exclude these multiples (Broggini and Snieder, 2012; Broggini et al., 2012; Wapenaar et al., 2013, 2014; Slob et al., 2014). In addition to the retrieved Green’s function, the Marchenko equations also give us the one-way focusing functions. These functions are outputs from the Marchenko equations that exist at the acquisition level ∂D0 (the acquisition surface) and focus on an arbitrary depth level ∂Di at t 1⁄4 0 (the time is equal to zero). The focusing functions are auxiliary wavefields that reside in a truncated medium that has the same material properties as the actual inhomogeneous medium between ∂D0 and ∂Di and that is homogeneous above ∂D0 and reflection-free below ∂Di (Slob et al., 2014). Therefore, the boundary conditions on ∂D0 and ∂Di in the truncated medium, in which the focusing function exists, are reflection free (see Figure 1). Our algorithm moves the sources of the Green’s function retrieved by the Marchenko equations from the surface into the subsurface at a virtual point with the help of the focusing function. In this paper, the spatial coordinates are defined by their horizontal and depth components; for instance, x0 1⁄4 ðxH; x3;0Þ, where xH stands for the horizontal coordinates at a depth x3 at the datum ∂D0, whereas xi 1⁄4 ðxH; x3;iÞ at the datum ∂Di. Superscript (þ) refers to downgoing waves and (−) refers to upgoing waves at the observation point x. For Green’s functions labeled as G , the superscript refers either to downgoing waves (þ) or to upgoing waves (−) at the observation point x. In addition, wavefield quantity with a subscript zero (e.g., R0) indicates that no free surface is present. One-way reciprocity theorems of the convolution and correlation type (equations 1 and 2) are used to relate upand downgoing fields at arbitrary depth levels to each other in different wave states (Wapenaar and Grimbergen, 1996). The one-way reciprocity theorems (for pressure-normalized wavefields) of the convolution and correlation type are Z ∂D0 ρ−1ðxÞ1⁄2ðpA Þ∂3pB þ p−Að∂3pB Þ dx0 1⁄4