Marchenko Theory, Algorithms, and Applications: A Review

Abstract

Seismic reflection data collected at the Earth's surface carry information from every depth in the subsurface, including the internal multiples that conventional migration treats as noise or artefacts. The Marchenko method retrieves subsurface Green's functions directly from these single-sided measurements. The only model information it requires is a smooth macro-velocity model, used to estimate the direct-wave traveltime from the surface to a virtual point at depth. The method originates in one-dimensional quantum mechanical inverse scattering and has been generalised over the past two decades to three-dimensional acoustic media using acoustic reciprocity theorems, and to elastic and electromagnetic media using the corresponding elastodynamic and Maxwell reciprocity theorems. This review covers the classical Gelfand--Levitan--Marchenko equation and its derivation through the Schrodinger scattering formalism, the seismic interferometry results that motivated the multidimensional extension, and the coupled Marchenko equations for wave focusing in higher dimensions. On the application side it treats Marchenko redatuming, single- and double-sided imaging conditions, internal multiple elimination, target-oriented processing, and full-waveform inversion. Extensions to dissipative media, elastodynamics, electromagnetic fields, and plane-wave acquisition are also covered, along with practical topics such as least-squares implementations, GPU acceleration, compressive-sensing acquisition, and recent machine-learning approaches to predicting the focusing functions. Field-data results from marine, land, sub-salt, and time-lapse monitoring settings are summarised. The review closes with an assessment of current limitations and open problems, among them elastic multi-component extensions, joint inversion, and neural-network accelerators.