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Abstract:

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.

Registro:

Documento: Artículo
Título:Determination of the bottom deformation from space- and time-resolved water wave measurements
Autor:Cobelli, P.J.; Petitjeans, P.; Maurel, A.; Pagneux, V.
Filiación:Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires and IFIBA, CONICET, Ciudad Universitaria, Buenos Aires, 1428, Argentina
Laboratoire de Physique et Mécanique des Milieux Hétérogènes, UMR CNRS 7636, Ecole Supérieure de Physique et de Chimie Industrielles, ESPCI-ParisTech, 10 rue Vauquelin, Paris Cedex 5, 75231, France
Institut Langevin, UMR CNRS 7587, Paris, France
Laboratoire D'Acoustique de L'Université du Maine, UMR CNRS 6613, Avenue Olivier Messiaen, Le Mans Cedex 9, 72085, France
Palabras clave:surface gravity waves; waves/free-surface flows; Deformation; Gravity waves; Phase interfaces; Water waves; Air liquid interfaces; Free-surface dynamics; Laboratory experiments; Regularization parameters; Regularization schemes; Surface gravity waves; Water wave measurements; Waves/free-surface flows; Inverse problems; free surface flow; gravity wave; inverse problem; surface energy; water wave
Año:2018
Volumen:835
Página de inicio:301
Página de fin:326
DOI: http://dx.doi.org/10.1017/jfm.2017.741
Título revista:Journal of Fluid Mechanics
Título revista abreviado:J. Fluid Mech.
ISSN:00221120
CODEN:JFLSA
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00221120_v835_n_p301_Cobelli

Referencias:

  • 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

Citas:

---------- APA ----------
Cobelli, P.J., Petitjeans, P., Maurel, A. & Pagneux, V. (2018) . Determination of the bottom deformation from space- and time-resolved water wave measurements. Journal of Fluid Mechanics, 835, 301-326.
http://dx.doi.org/10.1017/jfm.2017.741
---------- CHICAGO ----------
Cobelli, P.J., Petitjeans, P., Maurel, A., Pagneux, V. "Determination of the bottom deformation from space- and time-resolved water wave measurements" . Journal of Fluid Mechanics 835 (2018) : 301-326.
http://dx.doi.org/10.1017/jfm.2017.741
---------- MLA ----------
Cobelli, P.J., Petitjeans, P., Maurel, A., Pagneux, V. "Determination of the bottom deformation from space- and time-resolved water wave measurements" . Journal of Fluid Mechanics, vol. 835, 2018, pp. 301-326.
http://dx.doi.org/10.1017/jfm.2017.741
---------- VANCOUVER ----------
Cobelli, P.J., Petitjeans, P., Maurel, A., Pagneux, V. Determination of the bottom deformation from space- and time-resolved water wave measurements. J. Fluid Mech. 2018;835:301-326.
http://dx.doi.org/10.1017/jfm.2017.741