Short?term statistics of waves observed in deep water

The short?term statistics of 10 million individual waves observed with buoys in deep water have been investigated, corrected for a sample?rate bias, and normalized with the standard deviation of the surface elevation (the range of normalized wave heights is 0 < H < 10). The observed normalized trough depths are found to be Rayleigh distributed with near?perfect scaling. The normalized crest heights are also Rayleigh distributed but 3% higher than given by the conventional Rayleigh distribution. The observed normalized wave heights are not well predicted by the conventional Rayleigh distribution (overprediction by 9.5% on average), but they are very well predicted by Rayleigh?like distributions obtained from linear theories and by an empirical Weibull distribution (errors <1.5%). These linear theories also properly predict the observed monotonic variation of the normalized wave heights with the (de?)correlation between crest height and trough depth. The theoretical Rayleigh?like distributions may therefore be preferred over the empirical Weibull distribution and certainly over the conventional Rayleigh distribution. The values of the observed expected maximum wave height (normalized) as a function of duration are consistent with these findings. To inspect nonlinear effects, the buoy observations were supplemented with 10,000 waves observed with laser altimeters mounted on a fixed platform (0 < H< 7). The (normalized) crest heights thus observed are typically 5% higher than those observed with the buoys, whereas the (normalized) trough depths are typically 12% shallower. The distribution of the normalized wave heights thus observed is practically identical to the distribution observed with the buoys. These findings suggest that crest heights and trough depths are affected by nonlinear effects, but wave heights are not. One wave in our buoy observations may qualify as a freak wave