The adequacy or otherwise of various existing theoretical models for surface elevation, wave height, period and wave group statistics are tested against an extensive data set from the BP Forties field. Discussion then moves to the description of the long-term wave climate and the modelling of individual storm events. Some useful extensions to the presently available models are described, covering the joint probabilities of Hs and Tz and the prediction of storm duration and intensity.
It is common practice to differentiate between the properties of wave climates and random seas as judged (a) in the short term, over durations of 20minutes to three hours when statistical properties of the sea are assumed sensibly constant and (b) in the long-term, when non-stationary, seasonal and annual variability is central to the prediction of weather windows, fatigue environments and extreme design wave. In the short term, the work of Rice, Longuest-Higgins and many others established the properties of a linear random wave theory with a Gaussian distributed surface elevation spectra and the well known Rayleigh wave height distribution for zero-crossing waves. The analysis of long-term data has necessarily been more reliant upon empirical distributions. Ochi(1982) provides a most useful review of both short- and long-term models. The programme of probabilistic description of random seas mainly using surface elevation (¿) data collected by a Waverider buoy at the BP Forties field during six particularly severe storms, and data from Forties and elsewhere, spanning several years, in terms of summary statistics such as significant wave height (Hs and average zero-crossing period (Tz). It has been shown (James, 1986) that non-linearities in wave records from floating measurement systems can be suppressed, or at least attenuated, by the horizontal motion of the buoy. Therefore, the data used in the first section of this discussion has been drawn from one of the fixed instruments mounted on the Forties FB platform.
It has been commonly noted that there are consistent departures from the symmetrical Gaussian distribution of surface elevation in terms of positive skewness coefficients which may be related to finite amplitude, non-linear effects accentuating crest heights against trough depths. Variations of the Gram-Charlier perturbation on a Gaussian distribution have been proposed by Longuet-Higgins (1963) and Bitner (1980). Normalising surface elevation as y= ¿/¿', Longuet-Higgins proposed the probability density function (pdf) of y as
Where HJ(y) is the Jth Hermite polynomial, and¿, is µJ/s¿J, with µJ being the Jth central moment of the surface elevation:
Perhaps of more interest than the instantaneous surface elevation, the cumulative distribution function (cdf) of those crests (ac) and troughs (at) which constitute individual up-crossing wave heights (H=ac + at) can be approximated on the basis of narrow-bonded seas for which ¿ and ¿ (= d¿/dt) are statistically independent by
Where p(ac is the pdf of surface elevation form equation (1) and ¿m is the mode (mpv) of the pdf Figure 1 shows the distribution of normalized crests.