The elastodynamic Marchenko method removes overburden interactions obscuring the target information. This method either relies on separability of the so-called focusing and Green's functions or requires an accurate initial estimate of the focusing and Green's function overlap. Hitherto, F1- and G-+ have been assumed separable, whereas F1+ and (G-)* share an unavoidable overlap, which has been considered understood but hard to predict without knowing the model. However, velocity differences between P- and S-waves cause so far unexplored fundamental challenges for elastodynamic Marchenko autofocusing. These challenges are analysed for horizontally-layered media. First, the F1-/G-+ separability assumption can be violated depending on the medium, the redatuming depth and the angle of incidence. Second, the initial estimate of the said unavoidable overlap can be even more complicated than originally thought, including some of the internal multiples. We propose a strategy where we trade-off this sophisticated initial estimate with a trivial one at the cost of a more restrictive F1-/G-+ separability assumption, or at the cost of introducing an overlap between F1- and G-+ instead. The proposed method finds the desired solutions convolved by an unknown matrix which we can hope to remove by exploiting energy conservation and minimum-phase properties of the focusing functions.

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We investigate the behaviour of two non-linearly coupled flexural modes of a doubly clamped suspended beam (nanomechanical resonator). One of the modes is externally driven. We demonstrate that classically, the behavior of the non-driven mode is reminiscent of that of a parametrically driven linear oscillator: it exhibits a threshold behavior, with the amplitude of this mode below the threshold being exactly zero. Quantum-mechanically, we were able to access the dynamics of this mode below the classical parametric threshold. We show that whereas the mean displacement of this mode is still zero, the mean squared displacement is finite and at the threshold corresponds to the occupation number of 1/2. This finite displacement of the non-driven mode can serve as an experimentally verifiable quantum signature of quantum motion.@en