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An inverse measurement of the sudden underwater movement of the sea-floor by using the time-history record of the water-wave elevation. (English) Zbl 1231.86014
Summary: An inverse problem is formulated for (indirectly) measuring the sudden movement of the sea-floor by using a time-history record of the resulting water-wave elevation. The measurement problem is shown to have a unique solution, which can correspond to the real physical wave-source of the sudden movement of the sea-floor. However, the problem formulated herein is concerned with a Fredholm integral equation of the first kind. This leads to a peculiar-and unwelcome-phenomenon involving numerical instability in the solution of the integral equation because the solution of the first-kind integral equation lacks the stability property. Topologically, we are faced with a completely different mathematical solution-structure that is ill-posed in the sense of stability compared to the usual, well-posed problems. A solution-stability property is artificially inserted into the formulated (measurement) problem to find a stabilized solution: this is realized by introducing Landweber-Fridman’s regularization method in this study. The workability of the suggested measurement is investigated through a numerical experiment.
MSC:
86A22Inverse problems in geophysics
35R30Inverse problems for PDE
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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