| Summary: | A new long-term stochastic model, recently introduced by the same authors (Athanassoulis et al., 1992) is applied to the calculation of the probability structure of random quantities accumulated over long-term periods (long-term cumulative quantities). The underlying process (e.g., the sea-surface elevation, or a structural response of a ship) is modelled, in the long time, as a two-level (doubly) stochastic process, by distinguishing between the fast-time scale, in which the fluctuations are considered stationary, and the slow-time scale, in which the corresponding spectral characteristics are slowly evolving. As a first approximation, the time series of spectral parameters is given the structure of a renewal process, whose interarrival times are the durations of successive sea states. Then, long-term cumulative quantities can be considered as the 'accumulated cost' of a renewal-reward process, and their probability distribution is obtained in terms of the joint statistics of sea-state duration and spectral parameters, using a central limit theorem for renewal-reward processes, The limiting distribution is Gaussian with a mean value equal to that predicted by Battjes (1970). A new formula for predicting the variance is given in this paper. The long-term number of waves (cycles) having amplitude greater than a threshold value u, denoted by Mu, is treated as an example. Numerical results are presented for a site in the Central North Atlantic, based on 20-year hindcast data. The statistics of sea-state duration and spectral parameters is first obtained and discussed. Then, using these results, the probability distribution of Mu is estimated. It is found that the variation coefficient of Mu may be as great as 0.60, a fact indicating that Mu might not be adequately represented by its mean value. More elaborated models incorporating the dependence between successive sea states are expected to improve the prediction of the variance, without affecting the asymptotic normality of Mu.
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