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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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

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

Quantum computing could be a potential game-changer in industry sectors relying on the efficient solutions of large-scale global optimization problems. Exploration geoscience, is full of optimization problems and hence is a good candidate for application of quantum computing. It was recently suggested that quantum annealing, a form of adiabatic quantum computer, is a much better suited quantum computing platform for optimization problems than gate-based quantum computing. In this work, we show how the residual statics estimation problem can be solved on the quantum annealer and present our first results obtained on a quantum computer.

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Current seismic imaging methods require data that is free of multiple reflections, which is why a range of multiple-removal algorithms have been developed. However, state-of-the-art algorithms for internal multiple removal are based on single event identification. They fail in the presence of finely layered (sub-wavelength) media which cause short-period internal multiple reflections, since these cannot be resolved individually. We present a method for seismic short-period internal multiple removal for 2D and 3D media as an extension of recent 1D work, based on the Marchenko theory. If we can separate the medium into a horizontally layered overburden and an arbitrary complex underburden we are able to remove the overburden-related effects of short-period (and long-period) internal multiples. We do not require an impedance model but achieve this with a smooth background velocity model, similar to what is used in other multiple removal algorithms. We give a 2D numerical example with a finely layered horizontal overburden and a laterally inhomogeneous underburden. Comparison of the image derived with our augmented Marchenko scheme with a conventional Marchenko image results in a considerable uplift. This is the first time that short-period internal multiples are removed correctly from 2D seismic data in a purely data-driven way.

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