Separable space

Reflection (computer programming)

Algorithm

Inverse scattering problem

Monotonic function

Applied mathematics

Inverse

Computer science

Mathematical optimization

Geology

Paleontology

Mathematics

Multiple

Mathematical analysis

Geometry

Series (stratigraphy)

Programming language

Inverse problem

Arithmetic

Geophysics

The reflection response of strongly scattering media often contains complicated interferences between primaries and (internal) multiples, which can lead to imaging artifacts unless handled correctly. Internal multiples can be kinematically predicted, for example by the Jakubowicz method or by the inverse scattering series (ISS), as long as monotonicity, that is, “correct” temporal event ordering, is obeyed. Alternatively, the (conventional) Marchenko method removes all overburden-related wavefield interactions by formulating an inverse problem that can be solved if the Green’s and the so-called focusing functions are separable in the time domain, except for an overlap that must be predicted. For acoustic waves, the assumptions of the aforementioned methods are often satisfied within the recording regimes used for seismic imaging. However, elastic media support wave propagation via coupled modes that travel with distinct velocities. Compared to the acoustic case, not only does the multiple issue become significantly more severe, but also violation of monotonicity becomes much more likely. By quantifying the assumptions of the conventional Marchenko method and the ISS, unexpected similarities as well as differences between the requirements of the two methods come to light. Our analysis demonstrates that the conventional Marchenko method relies on a weaker form of monotonicity. However, this advantage must be compensated by providing more prior information, which in the elastic case is an outstanding challenge. Rewriting, or remixing, the conventional Marchenko scheme removes the need for prior information but leads to a stricter monotonicity condition, which is now almost as strict as for the ISS. Finally, we introduce two strategies on how the remixed Marchenko solutions can be used for imperfect, but achievable, demultiple purposes.

Comparison of monotonicity challenges encountered by the inverse scattering series and the Marchenko demultiple method for elastic waves