Relationship between one-sided and two-sided Green’s function representations

The Green’s function, defined as the response recorded at the acquisition surface for a source located in the interior of the subsurface, is a combination of the downgoing and upgoing wave fields needed to reconstruct an image of the discontinuities inside the earth. Two-sided Green’s function representations require measurement on the full boundary enclosing the domain of interest and allow us to retrieve the Green’s function originating from any location inside the medium. Practical constraints usually prevent the placement of receivers at depth inside the earth; hence standard imaging techniques need to apply approximations to two-sided Green’s function representations to construct an image of the subsurface. Recently-developed one-sided representations need data acquired on only one side of the boundary enclosing the region of interest. They present a practical advantage because they remove the physical constraints of having receivers at depth; hence one-sided representations do not require the approximations needed by two-sided techniques. In this paper, we show the connection between one-sided and two-sided Green’s function representations.