On the Joint Distribution of Excursion Duration and Amplitude of a Narrow-Band Gaussian Process

The probability density of crest amplitude and duration that exceeds a given level is used in many theoretical and practical problems in engineering that are subjected to fluctuating loads such as wind and wave loads. 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 stationary narrow-band Gaussian process. A density function is suggested that has the salient feature to depend only on the three lowest spectral moments $m_{0}, m_{1}, and~m_{2}$ and desired level of exceedance, $H$ . It does not require any condition on the autocorrelation function. This paper shows how increase in $H$ , increases the correlation between excursion periods and amplitude. This paper also shows that how the accuracy of the proposed joint distribution relates to spectral width parameter, $\nu $ , and that accuracy increases for higher levels of $H$ , especially for a spectrum describing a physical phenomenon such as a sea state spectrum. It was demonstrated that the marginal distribution of amplitude is Rayleigh distributed, as expected, and that the marginal distribution of excursion duration works for asymptotic and non-asymptotic levels. Results demonstrate that the established distribution fits well with ideal narrow-band Gaussian processes as well as the sea states at three European sites —in the Atlantic Ocean and the North Sea. The suggested model is found to be a good replacement for the existing empirical distributions.