Minimum-phase properties are well-understood for scalar functions where they can be used as physical constraint for phase reconstruction. Existing scalar applications of the latter in geophysics include, for example the reconstruction of transmission from acoustic reflection data, or multiple elimination via the augmented acoustic Marchenko method. We review scalar minimum-phase reconstruction via the conventional Kolmogorov relation, as well as a less-known factorization method. Motivated to solve practice-relevant problems beyond the scalar case, we investigate (1) the properties and (2) the reconstruction of minimum-phase matrix functions. We consider a simple but non-trivial case of 2 × 2 matrix response functions associated with elastodynamic wavefields. Compared to the scalar acoustic case, matrix functions possess additional freedoms. Nonetheless, the minimum-phase property is still defined via a scalar function, that is a matrix possesses a minimum-phase property if its determinant does. We review and modify a matrix factorization method such that it can accurately reconstruct a 2 × 2 minimum-phase matrix function related to the elastodynamic Marchenko method. However, the reconstruction is limited to cases with sufficiently small differences between P- and S-wave traveltimes, which we illustrate with a synthetic example. Moreover, we show that the minimum-phase reconstruction method by factorization shares similarities with the Marchenko method in terms of the algorithm and its limitations. Our results reveal so-far unexplored matrix properties of geophysical responses that open the door towards novel data processing tools. Last but not least, it appears that minimum-phase matrix functions possess additional, still-hidden properties that remain to be exploited, for example for phase reconstruction.

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Many seismic imaging methods use wavefield extrapolation operators to redatum sources and receivers from the surface into the subsurface. We discuss wavefield extrapolation operators that account for internal multiple reflections, in particular propagator matrices, transfer matrices and Marchenko focusing functions. A propagator matrix is a square matrix that 'propagates' a wavefield vector from one depth level to another. It accounts for primaries and multiples and holds for propagating and evanescent waves. A Marchenko focusing function is a wavefield that focuses at a designated point in space at zero time. Marchenko focusing functions are useful for retrieving the wavefield inside a heterogeneous medium from the reflection response at its surface. By expressing these focusing functions in terms of the propagator matrix, the usual approximations (such as ignoring evanescent waves) are avoided. While a propagator matrix acts on the full wavefield vector, a transfer matrix (according to the definition used in this paper) 'transfers' a decomposed wavefield vector (containing downgoing and upgoing waves) from one depth level to another. It can be expressed in terms of decomposed Marchenko focusing functions. We present propagator matrices, transfer matrices and Marchenko focusing functions in a consistent way and discuss their mutual relations. In the main text we consider the acoustic situation and in the appendices we discuss other wave phenomena. Understanding these mutual connections may lead to new developments of Marchenko theory and its applications in wavefield focusing, Green's function retrieval and imaging.

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Standard Marchenko redatuming and imaging schemes neglect evanescent waves and are based on the assumption that decomposition into downgoing and upgoing waves is possible in the subsurface. Recently we have shown that propagator matrices, which circumvent these assumptions, can be expressed in terms of Marchenko focusing functions. In this paper we generalize the relation between the propagator matrix and the Marchenko focusing functions for a 3D inhomogeneous dissipative medium. Moreover, for the same type of medium we discuss a relation between the transfer matrix and the Marchenko focusing functions.@en

Marchenko redatuming retrieves Green’s functions inside an unknown medium, by solving a set of coupled Marchenko equations, which are derived from an under-determined system of equation and two temporal truncations. To constrain the problem, two assumptions are made, which hold reasonably well for acoustic, but not for elastodynamic waves. First, an early part of the inverse transmission field is needed which can be estimated for sufficiently-simple acoustic cases, but remains hard to predict for elastic media without detailed overburden knowledge. Secondly, the scheme assumes temporal separability of up-going focusing and Green’s functions, which holds for many acoustic media but easily fails in presence of elastic effects. The impact of the failure to meet these assumptions is a somewhat controllable problem in 1.5D media. Independently, in acoustic media one can use a time-only focusing to retrieve focusing functions which collapse to a single plane wave below the overburden. We apply this approach to elastic data from a very complex almost 1.5D medium. The numerical example shows that the plane-wave approach can also be combined with mitigation of failure to satisfy the aforementioned assumptions and the result could lead to a high-fidelity internal multiple-free image.@en

Marchenko equation-based methods promise data-driven, true-amplitude internal multiple elimination. The method is exact in 1-D acoustic media, however it needs to be expanded to account for the presence of 2- and 3-D elastodynamic wave-field phenomena, such as compressional (P) to shear (S) mode conversions, total reflections or evanescent waves. Mastering high waveform-fidelity methods such as this, could further advance amplitude vs offset analysis and lead to improved reservoir characterization. This method-expansion may comprise of re-evaluating the underlying assumptions and/or appending the scheme with additional constraints (e.g. minimum phase). To do that, one may need to better understand the construction of the Marchenko equation solutions, the so-called focusing functions, in a mathematically simple and numerically stable fashion. The latter could be a challenge at large angles of incidence where the elastodynamic effects and evanescent waves start playing a dominant role. We demonstrate that the elastodynamic focusing functions are the bridge between the Marchenko equation theory and the transfer matrix formalism. Using the latter, we show how we can try to gain further insights into how time-reversal (correlations) behaves when either of the elastic modes becomes evanescent. We also show how this construction allows us to shed light on into the mathematical properties of elastodynamic inverse transmissions, which takes us a step closer towards understanding the elastodynamic minimum phase reconstruction.@en

The presence of evanescent modes and their impact on the Marchenko method has been until very recently a topic that received little attention. In this contribution we link the concept of the transfer matrix to the fields usually associated with the Marchenko method. Using that formalism, we introduce the concept of a path reversal - a generalization of time reversal for travelling waves, which also encompasses the evanescent modes. We take a first look at the implications the latter may have for the “standard” Marchenko method. It appears that the scheme should perform well, as long as one does not attempt to reconstruct evanescent modes inside the medium (at the target-overburden boundary).@en