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

Jang, T. S.; Han, S. L.; Kinoshita, T.

Wave Motion 47, No. 3, 146-155 (2010).

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:

86A22	Inverse problems in geophysics
35R30	Inverse problems for PDEs
76B15	Water waves, gravity waves; dispersion and scattering, nonlinear interaction

Keywords:

inverse problem; indirect-measurement; sudden movement of the sea-floor

References:

[1]	Mei, C. C., The Applied Dynamics of Ocean Surface Waves (1989), World Scientific: World Scientific London, pp. 20-35 · Zbl 0991.76003
[2]	Stoker, J. J., Water Waves (1992), Wiley-Interscience · Zbl 0812.76002
[3]	Whitham, G. B., Linear and Nonlinear Waves (1999), Wiley-Interscience · Zbl 0373.76001
[4]	Mehaute, B. L.; Wang, S., Water Waves Generated by Underwater Explosion (1996), World Scientific · Zbl 0857.76002
[5]	Sotiropoulos, D. A.; Achenbach, J. D., Crack characterization by an inverse scattering method, International Journal of Solids and Structures, 24, 165-175 (1988) · Zbl 0629.73085
[6]	Achenbach, J. D.; Viswanathan, K.; Norris, A., An inversion integral for crack-scattering data, Wave Motion, 1, 299-316 (1979) · Zbl 0418.73086
[7]	Cheney, M.; Isaacson, D., Inverse problems for a perturbed dissipative half-space, Inverse Problems, 11, 865-888 (1995) · Zbl 0842.35135
[8]	Mazzucato, A. L.; Rachele, L. V., On uniqueness in the inverse problem for transversely isotropic elastic media with a disjoint wave mode, Wave Motion, 44, 605-625 (2007) · Zbl 1231.35309
[9]	Janno, J.; Engelbrecht, J., Waves in microstructured solids: inverse problems, Wave Motion, 43, 1-11 (2005) · Zbl 1231.74197
[10]	Dominguez, N.; Gibiat, V.; Esquerre, Y., Time domain topological gradient and time reversal analogy: an inverse method for ultrasonic target detection, Wave Motion, 42, 31-52 (2005) · Zbl 1189.74070
[11]	Hellsten, H.; Maz’ya, V.; Vainberg, B., The spectrum of water waves produced by moving point sources, and a related inverse problem, Wave Motion, 38, 345-354 (2003) · Zbl 1163.74368
[12]	Kirsch, A., An Introduction to the Mathematical Theory of Inverse Problems (1996), Springer · Zbl 0865.35004
[13]	Tikhonov, A. N., Solution of incorrectly formulated problems and the regularization method, Soviet Mathematical Doklady, 4, 1035-1038 (1963) · Zbl 0141.11001
[14]	Kammerer, W. J.; Nashed, M. Z., Iterative methods for best approximate solutions of integral equations of the first and second kinds, Journal of Mathematicla Analysis and Applications, 40, 547-573 (1972) · Zbl 0246.45015
[15]	Landweber, L., An iteration formula for Fredholm integral equations of the first kind, American Journal of Mathematics, 73, 615-624 (1951) · Zbl 0043.10602
[16]	Hochstadt, H., Integral Equations (1973), Wiley Interscience Publication: Wiley Interscience Publication New York · Zbl 0137.08601
[17]	Groetsch, C. W., Inverse Problems in the Mathematical Sciences (1993), Vieweg · Zbl 1266.34019
[18]	Roman, P., Some Modern Mathematics for Physicists and Other Outsiders (1975), Pergamon Press Inc.: Pergamon Press Inc. New York, vol. 1, pp. 270-290
[19]	Jang, T. S.; Han, S. L., Application of Tikhonov’s regularization to an unstable two dimensional water waves: spectrum with compact support, Ships and Offshore Structures, 3, 41-47 (2008)
[20]	Jang, T. S.; Kinoshita, T., An ill-posed inverse problem of a wing with locally given velocity data and its analysis, Journal of Marine Science and Technology, 5, 16-20 (2000)
[21]	Jang, T. S.; Choi, H. S.; Kinoshita, T., Numerical experiments on an ill-posed inverse problem for a given velocity around a hydrofoil by iterative and noniterative regularizations, Journal of Marine Science and Technology, 5, 107-111 (2000)
[22]	Jang, T. S.; Choi, H. S.; Kinoshita, T., Solution of an unstable inverse problem: wave source evaluation from observation of velocity distribution, Journal of Marine Science and Technology, 5, 181-188 (2000)
[23]	Jang, T. S.; Kwon, S. H.; Kim, B. J., Solution of an unstable axisymmetric Cauchy-Poisson problem of dispersive water waves for a spectrum with compact support, Ocean Engineering, 34, 676-684 (2007)
[24]	Jang, T. S.; Sung, H. G.; Han, S. L.; Kwon, S. H., Inverse determination of the loading source of the infinite beam on elastic foundation, Journal of Mechanical Science and Technology, 22, 2350-2356 (2008)
[25]	Spiegel, M. R., Mathematical Handbook (1968), Mcgraw-hill
[26]	Hansen, P. C., Analysis of discrete ill-posed problems by means of the L-curve, SIAM Review, 34, 561-580 (1992) · Zbl 0770.65026