Distribution of wave crests in a non-Gaussian sea
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 , is often required. The exact form of the distribution is not known. For a Gaussian sea it is common practice to approximate the
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 coordinate. In addition we consider a unidirectional sea, where all waves travel along the axis with positive velocity. The linear sea , consisting of cosine waves, is given by where, for each elementary wave, denotes its complex valued amplitude, the angular
Mean upcrossing intensity
Initially suppose that is a stationary, zero mean Gaussian process. If the derivative exists then, for a fixed level , the expected number of times the process crosses in the upward direction, , is given by where is the joint density of . 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 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 .
We start by using the inversion formula to write as a two-dimensional inversion of . The inversion expression is used in place of , 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 is particularly simple, namely where is the peak frequency of the spectrum of the linear part . We select 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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Cited by (19)
Rogue wave statistics in (2+1) Gaussian seas I: Narrow-banded distribution
2020, Applied Ocean ResearchWave crest height distribution during the tropical cyclone period
2020, Ocean EngineeringCitation Excerpt :As the wave propagates, the profile becomes asymmetric with higher crests together with shallower and flatter troughs due to the interaction between individual sine waves. Butler et al. (2009) show that the linear wave model is often simplistic and leads to errors in the predicted crest height of about 10–20%. The theoretical derivation of the distribution of crest height is a tricky problem, especially when non-linearities have to be considered.
Analysis of the extreme wave elevation due to second-order diffraction around a vertical cylinder
2019, Applied Ocean ResearchCitation Excerpt :For a narrowband Gaussian process, it is known that the crests follow a Rayleigh distribution; however, the crests of non-Gaussian waves are non-Rayleigh. Various researchers [9,25–27] have proposed corrections to the Rayleigh distribution for nonlinear random waves. Estimating the extreme value based on the distribution of peak values relies on the assumption that the peaks are statistically independent, which is not strictly correct.
Prediction of structural response of naval vessels based on available structural health monitoring data
2016, Ocean EngineeringCitation Excerpt :Time-history responses for unobserved cells are then readily available to be used in cycling counting methods. The elevation of the sea surface is typically considered Gaussian (Butler et al., 2009). However, for larger waves, the shape of the wave deviates from a simple sinusoid and becomes cnoidal or otherwise Non-Gaussian (Osborne, 2010).
Nonlinear crest distribution for shallow water Stokes waves
2016, Applied Ocean ResearchCitation Excerpt :It should be noted that in the existing literature the Rice's logic has been already extended to model distributions of nonlinear asymmetric waves, including shallow water waves (see, e. g. Machado and Rychlik [31]; Butler et al. [34])). However, the approach in the present paper for the calculation of nonlinear crest distribution is very much different from the existing approaches as used in Machado and Rychlik [31] and in Butler et al. [34]. A key procedure in the present approach is to use a deterministic and memoryless functional transformation to relate a stationary zero mean variance one Gaussian process to the original non-Gaussian process.
Recent developments of ocean environmental description with focus on uncertainties
2014, Ocean EngineeringCitation Excerpt :It is also a conservative estimate that fits well to the tail of the distribution. A new approach on evaluation of the Rice formula for second order wave models, based on a saddle point approximation, was presented in Butler et al. (2009). Further Laplace distributed processes have been used to model waves at fixed location, see Åberg et al. (2009).