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Slepian models for the stochastic shape of individual Lagrange sea waves

Part of: Geophysics (general) Stochastic processes Special processes Geophysics

Published online by Cambridge University Press:  01 July 2016

Georg Lindgren
Georg Lindgren*
Affiliation:
Lund University
*
∗ Postal address: Department of Mathematical Statistics, Lund University, Box 118, SE-221 00 Lund, Sweden. Email address: georg.lindgren@matstat.lu.se
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Abstract

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Gaussian wave models have been successfully used since the early 1950s to describe the development of random sea waves, particularly as input to dynamic simulation of the safety of ships and offshore structures. A drawback of the Gaussian model is that it produces stochastically symmetric waves, which is an unrealistic feature and can lead to unconservative safety estimates. The Gaussian model describes the height of the sea surface at each point as a function of time and space. The Lagrange wave model describes the horizontal and vertical movements of individual water particles as functions of time and original location. This model is physically based, and a stochastic version has recently been advocated as a realistic model for asymmetric water waves. Since the stochastic Lagrange model treats both the vertical and the horizontal movements as Gaussian processes, it can be analysed using methods from the Gaussian theory. In this paper we present an analysis of the stochastic properties of the first-order stochastic Lagrange waves model, both as functions of time and as functions of space. A Slepian model for the description of the random shape of individual waves is also presented and analysed.

Keywords

Slepian modelcrossing theorywave steepnessGaussian field

MSC classification

Primary: 60G60: Random fields
Secondary: 60G70: Extreme value theory; extremal processes 60K40: Other physical applications of random processes 86A05: Hydrology, hydrography, oceanography
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 2 , June 2006 , pp. 430 - 450
Copyright
Copyright © Applied Probability Trust 2006 

References

Azaı¨s, J.-M., León, J. R. and Ortega, J. (2005). Geometrical characteristics of Gaussian sea waves. J. Appl. Prob. 42, 407–425.Google Scholar
Baxevani, A., Podgórski, K. and Rychlik, I. (2003). Velocities for moving random surfaces. Prob. Eng. Mechanics 18, 251–271.Google Scholar
Chang, M.-S. (1969). Mass transport in deep-water long-crested random gravity waves. J. Geophys. Res. 74, 1515–1536.CrossRefGoogle Scholar
Fouques, S., Krogstad, H. E. and Myrhaug, D. (2004). A second-order Lagrangian model for irregular ocean waves. In OMAE2004 (Proc. 23rd Internat. Conf. Offshore Mechanics and Arctic Engineering, Vancouver, 2004), American Society of Mechanical Engineers, New York.Google Scholar
Gjøsund, S. H. (2000). Kinematics in regular and irregular waves based on a Lagrangian formulation. , NTNU, Trondheim.Google Scholar
Gjøsund, S. H. (2003). A Lagrangian model for irregular waves and wave kinematics. J. Offshore Mechanics Arctic Eng. 125, 94–102.Google Scholar
Hasselmann, K. et al. (1973). Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP). Deustche Hydrogr. Z. Suppl. A 12, 1–95.Google Scholar
Kinsman, B. (1965). Wind Waves: Their Generation and Propagation on the Ocean Surface. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.Google Scholar
Lindgren, G. (1970). Some properties of a normal process near a local maximum. Ann. Math. Statist. 41, 1870–1883.Google Scholar
Lindgren, G. and Broberg, B. K. (2004). Cycle range distributions for Gaussian processes – exact and approximative results. Extremes 7, 69–89.CrossRefGoogle Scholar
Lindgren, G. and Rychlik, I. (1991). Slepian models and regression approximations in crossing and extreme value theory. Internat. Statist. Rev. 59, 195–225.CrossRefGoogle Scholar
Longuet-Higgins, M. S. (1957). The statistical analysis of a random, moving surface. Phil. Trans. R. Soc. London A 249, 321–387.Google Scholar
Machado, U. E. B. (2002). Statistical analysis of non-Gaussian environmental loads and responses. , Lund University, Sweden.Google Scholar
Machado, U. and Rychlik, I. (2003). Wave statistics in non-linear random sea. Extremes 6, 125–146.CrossRefGoogle Scholar
Miche, M. (1944). Mouvements ondulatoire de la mer en profondeur constante ou décroissante. Forme limite de la houle lors de son déferlment. Application aux digues marine. Ann. Ponts Chaussées 1944, 25–78, 131–164, 270–292, 369–406.Google Scholar
Pierson, W. J. (1961). Models of random seas based on the Lagrangian equations of motion. Tech. Rep. Contr. Nonr-285(03), College of Engineering, New York University.CrossRefGoogle Scholar
Podgórski, K., Rychlik, I. and Sjö, E. (2000). Statistics for velocities of Gaussian waves. Internat. J. Offshore Polar Eng. 10, 91–98.Google Scholar
Rychlik, I., Johannesson, P. and Leadbetter, M. R. (1997). Modelling and statistical analysis of ocean-wave data using transformed Gaussian processes. Marine Struct. 10, 13–47.CrossRefGoogle Scholar
Slepian, D. (1962). On the zeros of Gaussian noise. In Proc. Symp. Time Ser. Anal., ed. Rosenblatt, M., John Wiley, New York.Google Scholar
Socquet-Juglard, H. et al. (2004). Spatial extremes, shapes of large waves, and Lagrangian models. In Proc. Rogue Waves 2004, eds. Olagnon, M. and Prevosto, M., IFREMER. Available at http://www.ifremer.fr/web-com/stw2004/rw/fullpapers/krogstad.pdf.Google Scholar
St. Denis, M. and Pierson, W. J. (1953). On the motions of ships in confused seas. Trans. Soc. Naval Architects Marine Eng. 61, 280–357.Google Scholar