Direct Greens function retrieval with internal multiples: An alternative to Marchenko focusing

Subsurface Green's functions provide crucial information for the seismic imaging and redatuming. A complete Green's functions containing the primary reflections and all orders of multiples can be utilized to mitigate artifacts and improve the resolution of seismic imaging. To fulfill this goal, some data-driven approaches using one-sided recorded data from the Earth's surface and a smooth migration velocity of the reference medium were developed. Among these approaches an iterative scheme was proposed using the multidimensional Marchenko equation based focusing functions. The iterative Marchenko approach is intrinsically designed to retrieve the coda of the focusing functions, which is supposed to handle all the internal multiples. The estimated focusing functions are then utilized to calculate the Green's functions by a crosscorrelation step. Inspired by the generalized internal multiple imaging (GIMI), we propose an approach that directly retrieves the Green's functions, instead of solving for the focusing functions. In the GIMI process, the reflection data are projected into the subsurface using the transmission information, followed by an interferometric step, which is similar to the multidimensional crosscorrelation of the Marchenko implementation. Thus, we derive a projected Marchenko equation from the relation between the Green's functions and the focusing functions, which reveals a clear connection to the GIMI. The new formulation offers an opportunity to solve for the Green's functions using an iterative scheme or by dealing with different orders of scattering, separately (a hierarchic approach). We introduce these two schemes and the corresponding adjoint operations, which enable us to adopt an optimization for data fitting. The basic performance of the two schemes are demonstrated on synthetic examples for the purpose of redatuming.