Three-Dimensional Single-Sided Marchenko Inverse Scattering, Data-Driven Focusing, Green’s Function Retrieval, and their Mutual Relations

Abstract

The one-dimensional Marchenko equation forms the basis for inverse scattering problems in which the scattering object is accessible from one side only. Here we derive a three-dimensional (3D) Marchenko equation which relates the single-sided reflection response of a 3D inhomogeneous medium to a field inside the medium. We show that this equation is solved by a 3D iterative data-driven focusing method, which yields the 3D Green's function with its virtual source inside the medium. The 3D single-sided Marchenko equation and its iterative solution method form the basis for imaging of 3D strongly scattering inhomogeneous media that are accessible from one side only.

On the role of multiples in Marchenko imaging

“…In other words, the result at t ¼ 0 (responsible for the image of r 4 ) comes entirely from the first arrivals in the down-and upgoing Green's functions at z ϵ 4 , indicated by the deepest green arrow in Figure 1. Because G þ ðz; z 0 ; tÞ for arbitrary z > z 0 can be written as a convolution of two minimum-phase functions (Wapenaar et al [2013], equation 7), its inverse is causal (advanced by t d ðzÞ); hence, the same reasoning as above also holds for the images of the other reflectors (indicated by the other green arrows in Figure 1). Finally, using the fact that the coda of the focusing function is causal, it can be shown in a similar way that the first arrival in G − ðz ϵ n ; z 0 ; tÞ (n ¼ 1, 2, 3, 4) comes, in turn (via equation 1), from the primary reflection of the nth reflector, i.e., Rðz 0 ; 2t d ðz n ÞÞ.…”

Single-sided Marchenko focusing of compressional and shear waves

“…Da Costa et al [20][21][22] investigate an elastodynamic extension of Ref. [12] and illustrate with numerical experiments the advantages and limitations of their approach. In a recent paper [23] we briefly introduce another approach to generalize the single-sided Marchenko method for elastodynamic waves and show that, at least in principle, all multiply scattered and mode-converted waves are properly taken into account.…”

“…Their method focuses an acoustic wave field onto the strongest scatterers in the medium. The 1D autofocusing method of Rose [7,8] and our 3D single-sided Marchenko focusing method [12,13] focus acoustic waves at any point inside a medium, thereby accounting for internal multiple scattering. The 3D method requires, apart from the measured reflection response of the medium, an estimate of the direct waves between the focal point and the acquisition surface.…”

“…In the following we only consider situations for which these equations hold. This is the case, for example, at finite horizontal distances in layered media with moderately curved interfaces [12,13], assuming a focal point not too close to the interface above it. Applying the window matrix W(x T ,x F ,t) to Eqs.…”

“…They obtained the Green's function of the unknown medium between the virtual source and the surface. A 3D extension of these concepts [12,13] has led to new imaging methodology for single-sided scalar reflection data [14,15].…”