On the joint distribution of excursion duration and amplitude of a narrow-band Gaussian process

The probability density of crest amplitude and of duration of exceeding a given level are used in many theoretical and practical problems in engineering. The joint density is essential for design of constructions that are subjected to waves and wind. The presently available joint distributions of amplitude and period are limited to excursion through a mean-level or to describe the asymptotic behavior of high level excursions. This paper extends the knowledge by presenting a theoretical derivation of probability of wave exceedance amplitude and duration, for a narrow-band Gaussian process. The suggested density function has the following properties: (1) it only depends on the three lowest spectral moments m0, m1, m2
and desired level of exceedance, H. It does not require any condition on the autocorrelation function; (2) by increasing H, correlation between excursion periods and amplitudes increases; (3) for a spectrum describing a physical phenomenon such as a sea state spectrum, the accuracy of the proposed approximation, for a given spectral width parameter ν, increases for higher level H. The paper shows that the marginal distribution of amplitude is compatible with the Rayleigh distribution, as expected, and that the marginal distribution of excursion duration works both for asymptotic and non-asymptotic cases. The suggested model is found to be a good replacement for the empirical distributions that are widely used. Results from simulations of narrow-band Gaussian processes, real sea states at three European sites—in the Atlantic ocean and in North
sea—are found to agree well with the established distribution.