Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-18T09:55:06.508Z Has data issue: false hasContentIssue false
  • English
  • Français

Geometrical characteristics of Gaussian sea waves

Part of: Special processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Jean-Marc Azaïs ,
José R. León  and
Joaquín Ortega
Jean-Marc Azaïs*
Affiliation:
Université Paul Sabatier
José R. León*
Affiliation:
Universidad Central de Venezuela
Joaquín Ortega*
Affiliation:
CIMAT and Universidad Central de Venezuela
*
∗Postal address: Laboratoire de Statistique et Probabilités, Université Paul Sabatier, 118, Route de Narbonne, 31062 Toulouse cedex, France. Email address: [email protected]
∗∗Postal address: Facultad de Ciencias, Universidad Central de Venezuela, Caracas 1020, Venezuela.
∗∗Postal address: Facultad de Ciencias, Universidad Central de Venezuela, Caracas 1020, Venezuela.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this work, we study some geometrical properties of a stationary Gaussian field modeling the sea surface, using the energy spectrum. We consider the length of a crest and the mean speed of contours, which can be expressed as integrals over level sets. We also give central limit theorems for some of these quantities, using chaos expansions.

Keywords

Gaussian processvelocity level curvecrestcentral limit theorem

MSC classification

Primary: 60G15: Gaussian processes
Secondary: 60K40: Other physical applications of random processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 2 , June 2005 , pp. 407 - 425
Copyright
© Applied Probability Trust 2005 

References

Adler, R. J. (1981). The Geometry of Random Fields. John Wiley, Chichester.Google Scholar
Azaïs, J.-M. and Wschebor, M. (2005). On the distribution of the maximum of a Gaussian field with d parameters. Ann. Appl. Prob. 15, 254–278.Google Scholar
Baxevani, A., Podgórski, K. and Rychlik, I. (2003). Velocities for random surfaces. Prob. Eng. Mechanics 18, 251–271.Google Scholar
Benzaquen, S. and Cabaña, E. M. (1982). The expected measure of the level set of a regular stationary Gaussian process. Pacific J. Math. 103, 9–16.Google Scholar
Berzin, C. and Wschebor, M. (1993). Approximation du temps local des surfaces gaussiennes. Prob. Theory Relat. Fields 96, 1–32.CrossRefGoogle Scholar
Cabaña, E. M. (1985). Esperanzas de integrales sobre conjuntos de nivel aleatorios. In Proc. 2nd Latin American Congr. Prob. Math. Statist. (Caracas, 1985), Bernoulli Society, Caracas, pp. 65–82.Google Scholar
Cabaña, E. M. (1987). Affine processes: a test of isotropy based on level sets. SIAM J. Appl. Math. 47, 886–891.Google Scholar
Cramér, H. and Leadbetter, M. R. (1967). Stationary and Related Stochastic Processes. John Wiley, Chichester.Google Scholar
Corrsin, S. (1955). A measure of the area of a homogeneous random surface in space. Quart. Appl. Math. 12, 404–408.Google Scholar
Federer, H. (1969). Geometric Measure Theory. Springer, New York.Google Scholar
Kratz, M. and León, J. R. (2001). Central limit theorems for non-smooth functionals related to crossings (via the Hermite polynomial expansion). J. Theoret. Prob. 14, 639–671.Google Scholar
Krée, P. and Soize, C. (1983). Mécanique Aléatoire. Dunod, Paris.Google 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
Ochi, M. (1998). Ocean Waves. Cambridge University Press.Google 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
Wschebor, M. (1982). Formule de Rice en dimension d. Z. Wahrscheinlichkeitsth. 60, 393–401.Google Scholar
Wschebor, M. (1985). Surfaces Aléatoires (Lecture Notes Math. 1147). Springer, New York.Google Scholar
Ylvisaker, D. (1968). A note on the absence of tangencies in Gaussian sample paths. Ann. Math. Statist. 39, 261–262.Google Scholar
Zähle, U. (1984). A general Rice formula, Palm measures and horizontal-window conditioning for random fields. Stoch. Process Appl. 17, 265–283.Google Scholar