Elsevier

Applied Ocean Research

Volume 31, Issue 1, February 2009, Pages 57-64
Applied Ocean Research

Distribution of wave crests in a non-Gaussian sea

https://doi.org/10.1016/j.apor.2009.05.001Get rights and content

Abstract

The sea elevation at a fixed point is modelled as a quadratic form of a vector valued Gaussian process with arbitrary mean. With this model, saddlepoint methods are used to approximate the mean upcrossing intensity with which the sea level crosses upwards at a certain height. This estimated intensity is further used to determine the probability distribution of wave crests. The use of saddlepoint technique is particularly important here because it can approximate the crest distribution without the need to perform simulations or use fitted distributions. Several numerical examples are given, including two with measured data. In the cases of real data, the results obtained with the saddlepoint technique are also compared with the results obtained with well known methods commonly used in the industry.

Section snippets

Introduction and motivation

Commonly the sea surface elevation at a fixed point is modelled as a Gaussian process which, during a limited period of time (1–3 h), can be considered stationary. The model is called a Gaussian sea, and the parameters that characterize its power spectrum are the sea state. In reliability analysis of ocean structures the distribution of crest height, denoted by Ac, is often required. The exact form of the distribution is not known. For a Gaussian sea it is common practice to approximate the Ac

Modelling of the sea surface

We begin with the linear sea model, which postulates that the sea surface is a sum of simple cosine waves. In this paper we consider only long crested sea, i.e. the surface does not depend on the y coordinate. In addition we consider a unidirectional sea, where all waves travel along the x axis with positive velocity. The linear sea ηl, consisting of N cosine waves, is given by ηl(x,t)=n=NNAn2ei(ωntκnx), where, for each elementary wave, An denotes its complex valued amplitude, ωn the angular

Mean upcrossing intensity

Initially suppose that η(t) is a stationary, zero mean Gaussian process. If the derivative η̇(t) exists then, for a fixed level u, the expected number of times the process η(t) crosses u in the upward direction, μ+(u), is given by μ+(u)=0+zfη(0),η̇(0)(u,z)dz, where fη(0),η̇(0)(u,z) is the joint density of η(0),η̇(0). The above classical result is called Rice’s formula; see Leadbetter et al. [8] for a proof. It is easy to check that inserting the Gaussian density of η(0),η̇(0) into (10) will

Saddlepoint methods

The saddlepoint approximation was first introduced by Daniels [10], [11] as a formula to approximate the probability density function from its cumulant generating function. We shall apply here a variant of the method which will allow us to obtain directly an approximation of μ+(u).

We start by using the inversion formula to write fη(0),η̇(0)(u,z) as a two-dimensional inversion of exp{K(s,t)}. The inversion expression is used in place of fη(0),η̇(0)(u,z), in the integrand of (10), to form a

Numerical examples

In this section we demonstrate the accuracy of the saddlepoint approximation for the crossing intensity for four examples.

The first example considers the narrow-band Stokes waves in deep water. This is a situation in which the transfer function E(ω,ω̃) is particularly simple, namely E(ωm,ωn)=0,E(ωm,ωn)=ωp22g,n,m=1,,N, where ωp is the peak frequency of the spectrum S(ω) of the linear part ηl(t). We select S(ω) to be a JONSWAP1

Conclusions

The paper has demonstrated that the crest distribution of the waves in the second-order random sea model, defined by (9), can be very accurately approximated by saddlepoint approximation. The saddlepoint formulas for the crossing intensity are explicit but contain higher-order derivatives of the cumulant generating function, which usually have to be computed numerically. Programs in MATLAB that compute the saddlepoint approximation of the crossing intensity are available. The proposed method

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