Palm distributions of wave characteristics in encountering seas

Palm distributions of wave characteristics in encountering seas

Sofia Aberg, Igor Rychlik, and M. Ross Leadbetter

Source: Ann. Appl. Probab. Volume 18, Number 3 (2008), 1059-1084.

Abstract

Distributions of wave characteristics of ocean waves, such as wave slope, waveheight or wavelength, are an important tool in a variety of oceanographic applications such as safety of ocean structures or in the study of ship stability, as will be the focus in this paper. We derive Palm distributions of several wave characteristics that can be related to steepness of waves for two different cases, namely for waves observed along a line at a fixed time point and for waves encountering a ship sailing on the ocean. The relation between the distributions obtained in the two cases is also given physical interpretation in terms of a “Doppler shift” that is related to the velocity of the ship and the velocities of the individual waves.

First Page: Show

Primary Subjects: 60G15

Secondary Subjects: 60K40

Keywords: Encountered waves; Gaussian process; level crossings; Palm distribution; Rice’s formula; wave velocity

Full-text: Open access

Screen Optimized PDF File (281 KB)

Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aoap/1211819794
Digital Object Identifier: doi:10.1214/07-AAP480
Mathematical Reviews number (MathSciNet): MR2418238
Zentralblatt MATH identifier: 1146.60032

back to Table of Contents

References

[1] Adler, R. J. (1981). The Geometry of Random Fields. Wiley, New York.

Mathematical Reviews (MathSciNet): MR611857

Zentralblatt MATH: 0478.60059

[2] Azaïs, J. M., León, J. R. and Ortega, J. (2005). Geometrical characteristics of Gaussian sea waves. J. Appl. Probab. 42 407–425.

[3] Azaïs, J. M. and Wschebor, M. (2005). On the distribution of the maximum of a Gaussian field with d parameters. Ann. Appl. Probab. 15 254–278.

[4] Baxevani, A., Podgórski, K. and Rychlik, I. (2003). Velocities for moving random surfaces. Probabilist. Eng. Mech. 18 251–271.

[5] Brillinger, D. R. (1972). On the number of solutions of systems of random equations. Ann. Math. Statist. 43 534–540.

Mathematical Reviews (MathSciNet): MR300329

Digital Object Identifier: doi:10.1214/aoms/1177692634

Project Euclid: euclid.aoms/1177692634

[6] Brodtkorb, P. A., Johannesson, P., Lindgren, G., Rychlik, I., Rydén, J. and Sjö, E. (2000). WAFO—a Matlab toolbox for analysis of random waves and loads. In Proceedings of the 10th International Offshore and Polar Engineering Conference Seattle 3 343–350.

[7] Cramér, H. and Leadbetter, M. R. (1967). Stationary and Related Stochastic Processes. Sample Path Properties and Their Applications. Wiley, New York.

Mathematical Reviews (MathSciNet): MR217860

[8] Kac, M. (1943). On the average number of real roots of a random algebraic equation. Bull. Amer. Math. Soc. 49 314–320.

Mathematical Reviews (MathSciNet): MR7812

Digital Object Identifier: doi:10.1090/S0002-9904-1943-07912-8

[9] Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.

Mathematical Reviews (MathSciNet): MR691492

Zentralblatt MATH: 0518.60021

[10] Leadbetter, M. R. and Spaniolo, G. V. (2004). Reflections on Rice’s formulae for level crossings—history, extensions and use. Festschrift in honour of Daryl Daley. Aust. N. Z. J. Stat. 46 173–180.

Mathematical Reviews (MathSciNet): MR2060961

[11] Lindgren, G. and Rychlik, I. (1991). Slepian models and regression approximations in crossing and extreme value theory. Internat. Statist. Rev. 59 195–225.

[12] Longuet-Higgins, M. S. (1957). The statistical analysis of a random, moving surface. Philos. Trans. Roy. Soc. London Ser. A 249 321–387.

Mathematical Reviews (MathSciNet): MR87257

Digital Object Identifier: doi:10.1098/rsta.1957.0002

JSTOR: links.jstor.org

[13] Mercadier, C. (2005). Extrema de processus stochastiques. Propriétés asymptotiques de tests d’hypothèses. Ph.D. thesis, Univ. Toulouse.

[14] Podgórski, K., Rychlik, I. and Sjö, E. (2000). Statistics for velocities of Gaussian waves. Int. J. Offshore Polar Engineering 10 91–98.

[15] Rice, S. (1944, 1945). The mathematical analysis of random noise. I, II. Bell Syst. Techn. J. 23, 24 282–332, 145–156.

Mathematical Reviews (MathSciNet): MR10932

[16] Rychlik, I., Aberg, S. and Leadbetter, M. R. (2007). Note on the intensity of encountered waves. Ocean Eng. 34 1561–1568.

[17] Sjö, E. (2001). Simultaneous distributions of space-time wave characteristics in a Gaussian sea. Extremes 4 263–288.

Mathematical Reviews (MathSciNet): MR1907064

Digital Object Identifier: doi:10.1023/A:1015211022147

Zentralblatt MATH: 1051.60055

[18] Zähle, U. (1984). A general Rice formula, Palm measures, and horizontal-window conditioning for random fields. Stochastic Process. Appl. 17 265–283.

Mathematical Reviews (MathSciNet): MR751206

Digital Object Identifier: doi:10.1016/0304-4149(84)90005-X

Zentralblatt MATH: 0541.60049

previous :: next