Physics-based approach to wave statistics and probability

Design criteria in ocean engineering, whether this is one in 50 years or one in 5000 years event, are hardly ever based on measurements, and rather on statistical distributions of relevant metocean properties. Of utmost interest is the tail of these distributions, that is rare events such as the highest waves with low probability. Engineers have long since realised that the superposition of linear waves with narrow-banded spectrum as depicted by the Rayleigh distribution underestimates the probability of extreme wave crests, and is not adequate for wave heights either, which is a critical shortcoming as far as the engineering design is concerned. Ongoing theoretical and experimental efforts have been under way for decades to address this issue. Here, we will concentrate on short-term statistics, i.e. probability of crests/heights of individual waves. Typical approach is to treat all possible waves in the ocean or at a particular location as a single ensemble for which some comprehensive solution can be found. The oceanographic knowledge, however, now indicates that no single and united comprehensive solution is possible. Probability distributions in different physical circumstances should be different, and by combining them together the inevitable scatter is introduced. The scatter and the accuracy will not improve by increasing the bulk data quality and quantity, and it hides the actual distribution of extreme events. The groups have to be separated and their probability distributions treated individually. The paper offers a review of physical conditions, from simple one-dimensional trains of free waves to realistic twodimensional wind-forced wave fields, in order to understand where different probability distributions can be expected. If the wave trains/fields in the wave records are stable, distributions for the second-order waves should serve well. If modulational instability is active, rare extreme events not predicted by the second-order theory should become possible. This depends on wave steepness, bandwidth and directionality. Mean steepness also defines the wave breaking and therefore the upper limit for wave heights in this group of conditions. Under hurricane-like circumstances, the instability gives way to direct wind forcing, and yet another statistics is to be expected.