| Medium imaging is one of the most important study areas both in ocean acoustics and seismic exploration, where it is crucially to retrieve the Green’s function of sound or seismic waves in the medium At present, there exist two methods to retrieve the Green’s function:Green’s function retrieval by correlation (GFRC), also named "seismic interferometry" in seismology, and Green’s function retrieval by single-sided focusing (GFRSSF). The GFRC method needs to pose a receiver at the "virtual source" position inside the media, in contrast, the GFRSSF method just needs to lay receivers on one surface of the medium and measure the reflection response. From this aspect, the GFRSSF method is more available and economical. However, the single-sided focusing (SSF), the key point of the GFGSSF method, still has some problems urgently need for studies. For instance, it is still not clear how the one-dimensional (1D) SSF works inside the medium; the strict theory of the 1D SSF, which is established in the framework of the completeness relationship and the Marchenko equation, has not been expanded to higher dimension case.This paper studies the 1D single-sided acoustic focusing in a Goupillaud layered model. Sound fields in the model can be represented straightforwardly using a travel-time diagram This provides an intuitive physical picture of how the incident waveform of a 1D discrete SSF counteracts unwanted scattered sound fields. A recursive algorithm has been proposed to construct the incident waveform according to this physical interpretation. The proposed method gives a recursive solution of the corresponding discrete Marchenko equation, which governs the 1D discrete SSF of sound in a Goupillaud layered model. Two numerical scattering experiments, with different layered structures, are simulated to verify the proposed method.This paper also studies the single-sided focusing theory of the 1D multi-channel Schrodinger equations (i.e. quantum waveguide) in the framework of the completeness relationship and the Marchenko equation. To avoid complicated mathematical analysis, both the threshold energies and the bound states are assumed not exist in this paper. A multi-channel autofocusing procedure is proposed to construct the incident waveform required for the SSF in a quantum waveguide, and two numerical experiments are simulated to verify the proposed method. What is more, the completeness relationship of the 1D multi-channel Schrodinger equations, which can be explained as the single-sided focusing of the wave function in a quantum waveguide, is proved in this paper by virtue of the Green’s theorem. |
