An inverse measurement of the sudden underwater movement of the sea-floor by using the time-history record of the water-wave elevation

Abstract 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.

[1]  P. Roman.  Some Modern Mathematics for Physicists and Other Outsiders: An Introduction to Algebra, Topology, and Functional Analysis , 2014 .

[2]  L. Chambers.  Linear and Nonlinear Waves , 2000, The Mathematical Gazette.

[3]  A. Kirsch.  An Introduction to the Mathematical Theory of Inverse Problems , 1996, Applied Mathematical Sciences.

[4]  C. W. Groetsch,et al.  Inverse Problems in the Mathematical Sciences , 1993 .

[5]  Vincent Gibiat,et al.  Time domain topological gradient and time reversal analogy: an inverse method for ultrasonic target detection , 2005 .

[6]  Hans Hellsten,et al.  The spectrum of water waves produced by moving point sources, and a related inverse problem , 2003 .

[7]  Takeshi Kinoshita,et al.  An ill-posed inverse problem of a wing with locally given velocity data and its analysis , 2000 .

[8]  Anna L. Mazzucato,et al.  On uniqueness in the inverse problem for transversely isotropic elastic media with a disjoint wave mode , 2007 .

[9]  Taek Soo Jang,et al.  Inverse determination of the loading source of the infinite beam on elastic foundation , 2008 .

[10]  M. Nashed,et al.  Iterative Methods for Best Approximate Solutions of Linear Integral Equations of the First and Second Kinds , 1972 .

[11]  Bong Ju Kim,et al.  Solution of an unstable axisymmetric Cauchy-Poisson problem of dispersive water waves for a spectrum with compact support , 2007 .

[12]  C. Mei.  The applied dynamics of ocean surface waves , 1983 .

[13]  Takeshi Kinoshita,et al.  Solution of an unstable inverse problem: wave source evaluation from observation of velocity distribution , 2000 .

[14]  Takeshi Kinoshita,et al.  Numerical experiments on an ill-posed inverse problem for a given velocity around a hydrofoil by iterative and noniterative regularizations , 2000 .

[15]  Taek Soo Jang,et al.  Application of Tikhonov's regularization to unstable water waves of the two-dimensional fluid flow: spectrum with compact support , 2008 .

[16]  Jan Drewes Achenbach,et al.  Crack characterization by an inverse scattering method , 1988 .

[17]  Per Christian Hansen,et al.  Analysis of Discrete Ill-Posed Problems by Means of the L-Curve , 1992, SIAM Rev..

[18]  Jan Drewes Achenbach,et al.  An inversion integral for crack-scattering data , 1979 .

[19]  David Isaacson,et al.  Inverse problems for a perturbed dissipative half-space , 1995 .

[20]  A Tikhonov,et al.  Solution of Incorrectly Formulated Problems and the Regularization Method , 1963 .

[21]  Shen Wang,et al.  Water Waves Generated By Underwater Explosion , 1996 .

[22]  Jaan Janno,et al.  Waves in microstructured solids: Inverse problems , 2005 .

[23]  L. Landweber.  An iteration formula for Fredholm integral equations of the first kind , 1951 .