Determination of the bottom deformation from space- and time-resolved water wave measurements
In this paper we study both theoretically and experimentally the inverse problem of indirectly measuring the shape of a localized bottom deformation with a non-instantaneous time evolution, from either an instantaneous global state (space-based inversion) or a local time-history record (time-based i...
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| Otros Autores: | , , |
| Formato: | Capítulo de libro |
| Lenguaje: | Inglés |
| Publicado: |
Cambridge University Press
2018
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| Acceso en línea: | Registro en Scopus DOI Handle Registro en la Biblioteca Digital |
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| Sumario: | In this paper we study both theoretically and experimentally the inverse problem of indirectly measuring the shape of a localized bottom deformation with a non-instantaneous time evolution, from either an instantaneous global state (space-based inversion) or a local time-history record (time-based inversion) of the free-surface evolution. Firstly, the mathematical inversion problem is explicitly defined and uniqueness of its solution is established. We then show that this problem is ill-posed in the sense of Hadamard, rendering its solution unstable. In order to overcome this difficulty, we introduce a regularization scheme as well as a strategy for choosing the optimal value of the associated regularization parameter. We then conduct a series of laboratory experiments in which an axisymmetric three-dimensional bottom deformation of controlled shape and time evolution is imposed on a layer of water of constant depth, initially at rest. The detailed evolution of the air-liquid interface is measured by means of a free-surface profilometry technique providing space- and time-resolved data. Based on these experimental data and employing our regularization scheme, we are able to show that it is indeed possible to reconstruct the seabed profile responsible for the linear free-surface dynamics either by space- or time-based inversions. Furthermore, we discuss the different relative advantages of each type of reconstruction, their associated errors and the limitations of the inverse determination. © 2017 Cambridge University Press. |
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| Bibliografía: | Abramowitz, M., Stegun, I.A., (1964) Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, Applied Mathematics Series, , Dover Arfken, G., (1985) Inverse Laplace Transformation, Mathematical Methods for Physicists, , Academic Press Braddock, R.D., Van Den Driessche, P., Peady, G.W., Tsunami generation (1973) J. Fluid Mech., 59, pp. 817-828 Calvetti, D., Morigi, S., Reichel, L., Sgallari, F., Tikhonov regularization and the L-curve for large discrete ill-posed problems (2000) J. Comput. Appl. Maths, 123 (1-2), pp. 423-446 Cobelli, P.J., Maurel, A., Pagneux, V., Petitjeans, P., Global measurement of water waves by Fourier transform profilometry (2009) Exp. Fluids, 46, pp. 1037-1047 Cobelli, P.J., Pagneux, V., Maurel, A., Petitjeans, P., Experimental observation of trapped modes in a water wave channel (2009) Europhys. Lett., 88, p. 20006 Cobelli, P.J., Pagneux, V., Maurel, A., Petitjeans, P., Experimental study on waterwave trapped modes (2011) J. Fluid Mech., 666, pp. 445-476 Cobelli, P., Przadka, A., Petitjeans, P., Lagubeau, G., Pagneux, V., Maurel, A., Different regimes for water wave turbulence (2011) Phys. Rev. Lett., 107, p. 214503 Conway, J.B., A course in operator theory (1999) American Mathematical Society Dutykh, D., Dias, F., Water waves generated by a moving bottom (2007) Tsunami and Nonlinear Waves, pp. 65-95. , (ed. A. Kundu), Springer Eldén, L., Algorithms for the regularization of ill-conditioned least squares problems (1977) BIT Numer. Maths, 17 (2), pp. 134-145 Engl, H.W., Hanke, M., Neubauer, A., (2000) Regularization of Inverse Problems, Mathematics and Its Applications, , Springer Golub, G.H., Heath, M., Wahba, G., Generalized cross-validation as a method for choosing a good ridge parameter (1979) Technometrics, 21 (2), pp. 215-223 Groetsch, C.W., The theory of tikhonov regularization for fredholm equations of the first kind (1984) Pitman Adv. Publ. Program. Research Notes in Mathematics Series. Pitman Hadamard, J., (2003) Lectures On Cauchy's Problem in Linear Partial Differential Equations, , Dover Phoenix Editions. Dover Hammack, J.L., A note on tsunamis: Their generation and propagation in an ocean of uniform depth (1973) J. Fluid Mech., 60, pp. 769-799 Hansen, P.C., Analysis of discrete ill-posed problems by means of the L-curve (1992) SIAM Rev., 34 (4), pp. 561-580 Hansen, P.C., Discrete inverse problems: Insight and algorithms, fundamentals of algorithms (2010) Society for Industrial and Applied Mathematics Hansen, P.C., Jensen, T.K., Rodriguez, G., An adaptive pruning algorithm for the discrete L-curve criterion (2007) J. Comput. Appl. Maths, 198 (2), pp. 483-492 Hansen, P.C., O'leary, D.P., The use of the L-curve in the regularization of discrete ill-posed problems (1993) SIAM J. Sci. Comput., 14, pp. 1487-1503 Jang, T.S., A method for simultaneous identification of the full nonlinear damping and the phase shift and amplitude of the external harmonic excitation in a forced nonlinear oscillator (2013) Comput. Struct., 120, pp. 77-85 Jang, T.S., Han, S.L., Application of Tikhonov's regularization to unstable water waves of the two-dimensional fluid flow: Spectrum with compact support (2008) Ships Offshore Struct., 3 (1), pp. 41-47 Jang, T.S., Han, S.L., Kinoshita, T., An inverse measurement of the sudden underwater movement of the sea-floor by using the time-history record of the water-wave elevation (2010) Wave Motion, 47, pp. 146-155 Jang, T.S., Sung, H.G., Park, J., A determination of an abrupt motion of the sea bottom by using snapshot data of water waves (2012) Math. Prob. Engng, 2012 (4), pp. 472-575 Kajiura, K., The leading wave of tsunami (1963) Bull. Earthq. Res. Inst., 41, pp. 535-571 Keller, J.B., Tsunamis: Water waves produced by earthquakes (1961) Proceedings of Tsunami Hydrodynamics Conference, 24, pp. 154-166. , Institute of Geophysics, University of Hawaii Kirsch, A., (2011) An Introduction to the Mathematical Theory of Inverse Problems, Applied Mathematical Sciences, , Springer Lagubeau, G., Fontelos, M.A., Josserand, C., Maurel, A., Pagneux, V., Petitjeans, P., Flower patterns in drop impact on thin liquid films (2010) Phys. Rev. Lett., 105 (18), p. 184503 Lamb, H., (1932) Hydrodynamics, , 6th edn. Cambridge University Press Lawson, C.L., Hanson, R.J., Solving linear least squares problems (1995) Society for Industrial and Applied Mathematics Goff, A.L.E., Cobelli, P., Lagubeau, G., Supershear Rayleigh waves at a soft interface (2013) Phys. Rev. Lett., 110, p. 236101 Maurel, A., Cobelli, P.J., Pagneux, V., Petitjeans, P., Experimental and theoretical inspection of the phase-to-height relation in Fourier transform profilometry (2009) Appl. Opt., 48 (2), pp. 380-392 Mei, C.C., Stiassnie, M., Yue, D.K.P., Theory and applications of ocean surface waves: Linear aspects, advanced series on ocean engineering (2005) World Scientific Murty, T.S., Aswathanarayana, U., Nirupama, N., (2007) The Indian Ocean Tsunami, , Taylor & Francis Przadka, A., Cabane, B., Pagneux, V., Maurel, A., Petitjeans, P., Fourier transform profilometry for water waves: How to achieve clean water attenuation with diffusive reflection at the water surface? (2012) Exp. Fluids, 52 (2), pp. 519-527 Shannon, C.E., Communication in the presence of noise (1949) Proc. IRE, 37 (1), pp. 10-21 Sneddon, I.N., (1951) Fourier Transforms, International Series in Pure and Applied Mathematics, , McGraw-Hill Taubin, G., Estimation of planar curves, surfaces and nonplanar space curves defined by implicit equations, with applications to edge and range image segmentation (1991) IEEE Trans. Pattern Anal. Mach. Intell., 13, pp. 1115-1138 Tikhonov, A.N., On the stability of inverse problems (1943) Dokl. Akad. Nauk SSSR, 35, pp. 195-198 Tikhonov, A.N., Solution of incorrectly formulated problems and the regularization method (1963) Sov. Maths, 4, pp. 1035-1038 Tikhonov, A.N., Arsenin, V.I.A., (1977) Solutions of Ill-Posed Problems, Scripta Series in Mathematics, , Winston Tricomi, F.G., (1985) Integral Equations, Pure and Applied Mathematics, 5. , Dover Vaughn, M.T., (2007) Introduction to Mathematical Physics, Physics Textbook, , Wiley Wazwaz, A.M., (2011) Linear and Nonlinear Integral Equations: Methods and Applications, , Higher Education Press Yosida, K., (1995) Functional Analysis, Classics in Mathematics, , Cambridge University Press |
| ISSN: | 00221120 |
| DOI: | 10.1017/jfm.2017.741 |