| Summary: | We are concerned with the quantum inverse scattering problem. The corresponding
Marchenko integral equation is solved by using the collocation method together with
piece-wise polynomials, namely, Hermite splines. The scarcity of experimental data
and the lack of phase information necessitate the generation of the input reflection coefficient by choosing a specific profile and then applying our method to reconstruct it.
Various aspects of the single and coupled channels inverse problem and details about
the numerical techniques employed are discussed.
We proceed to apply our approach to synthetic seismic reflection data. The transformation
of the classical one-dimensional wave equation for elastic displacement into a
Schr¨odinger-like equation is presented. As an application of our method, we consider
the synthetic reflection travel-time data for a layered substrate from which we recover
the seismic impedance of the medium. We also apply our approach to experimental
seismic reflection data collected from a deep water location in the North sea. The
reflectivity sequence and the relevant seismic wavelet are extracted from the seismic
reflection data by applying the statistical estimation procedure known as Markov Chain
Monte Carlo method to the problem of blind deconvolution. In order to implement the
Marchenko inversion method, the pure spike trains have been replaced by amplitudes
having a narrow bell-shaped form to facilitate the numerical solution of the Marchenko
integral equation from which the underlying seismic impedance profile of the medium
is obtained. === Physics === D.Phil.(Physics)
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