A method of calculation of wind wave height probability based on the significant wave height probability is described (Chalikov and Bulgakov, 2017). The method can also be used for estimation of the height of extreme waves of any given cumulative probability. The application of the method on the basis of long-term model data is presented. Examples of averaged annual and seasonal fields of extreme wave heights obtained using the above method are given. Areas where extreme waves can appear are shown.

The highest risks of economic and environmental damage for sea-based human activities, i.e. cargo shipments, fishery, oil production etc., are mostly connected with extreme weather conditions on the sea surface, among which strong storms are the foremost. It is especially difficult to predict emergency situations caused by extreme waves for those cases of sea-based activities which require people's long stay at sea or prolonged use of equipment in the ocean.

One of the methods to minimize possible risks is the use of climate data based
on long-term series of observations. At present there are archives
consisting of reanalysis data on surface waves based on wave forecasts
corrected by different methods, i.e. direct measurements using
accelerometers and GPS buoys, remote measurements by satellite-borne
altimetry and various types of radars. The main characteristic of wave fields
included in the archive is significant wave height

It is evident that knowledge on significant wave height is not sufficient to
evaluate real wave height for a given wave field. Extreme waves of the same
height can appear with different probability for different values of

The nature of freak waves was investigated analytically (Onorato et al., 2009) and numerically (Chalikov, 2009). Recently it was found that the statistical properties of trough-to-crest wave height are quite different from those of the wave height above mean level. Studies (Chalikov and Babanin, 2016; Chalikov, 2016, 2017) show that linear and non-linear statistics of extreme waves (defined as trough-to-crest waves) are identical not only for a broad spectrum but for one-dimensional wave fields too. This means that generation of a trough-to-crest extreme wave is the result of the simple superposition of linear modes, no matter how broad the spectrum is. This property is not found for the wave height above mean level. Thus, the statistical properties of trough-to-crest wave height can be investigated with linear modelling, just by generation of large ensembles of the superposition of linear modes with random phases and the spectrum prescribed. Thus the problem of trough-to-crest statistics becomes quite straightforward. Contrary to such an approach, investigation of the statistics of wave height above mean level remains a subject of non-linear wave theory. From the practical point of view, for floating objects the data on the full height (trough to crest) of a wave are more important. However, the data on probability of wave height above mean level are important for fixed-construction offshore platforms.

The theoretical probability distribution for wave crest height (or wave height above mean level) was suggested by Weibull (1951). Later it was studied on a basis of observational data in nature and wave channels (see review by Kharif et al., 2009). Extended data for estimation of probability of wave height can be obtained with integration of non-linear modes based on full equations for potential (irrotational) flow (Touboul and Kharif, 2010; Chalikov, 2009). Methods of probability calculations were considered in many papers (see, for example, Bitner-Gregersen and Toffoli, 2012; Dyachenko et al., 2016).

The most popular method of trough-to-crest wave height detection is based on a zero-crossing technique. A direct method is based on the use of moving windows; the method is applicable for both 1-D and 2-D cases.

Estimation of extreme waves today is mostly made by analysis of data of
significant wave height. Jiangxia (2018), analysing long-term data,
considered that an extreme wave is a wave exceeding two significant wave
heights. In Larsen et al. (2015) a long-term wave dataset was analysed using
the
spectral method, and it was shown that the spectrum of modelled significant wave
height (trough to crest) contained the energy for a frequency of more than

This paper is devoted to investigation of the statistics and geographical distribution of wave height above mean sea level.

In Chalikov and Bulgakov (2017) an algorithm for estimation of cumulative
probability of waves

The probability of a wave exceeding a specific height

The model

The approximation of

The results of the series of experiments were processed in the following
way: each wave field of surface height above mean level (

Wave heights (m) above mean level with a cumulative probability of 10

The above expression can be used for the interval

The spatial distribution of extreme wave probability was investigated, based
on Eq. (3) from Chalikov and Bulgakov (2017) together with the spatial
distribution of significant wave height from Chawla et al. (2013). In this
paper results of an application of this method are considered. We show
global fields of wave height with a cumulative probability of 10

Figure 1 shows an average annual field of wave heights with a cumulative
probability of 10

The distribution of annual-average significant wave height provided by the model (Chawla et al., 2013) is shown in Fig. 2. As seen, the maximum value in the field of annual-average significant wave height does not exceed 5 m (southern area of Indian and Pacific oceans), while the height of real extreme waves can reach 16 m in this area. The data in Fig. 1 have a more complicated structure, due, for example, to the periods with strong wind along trajectories of tropical storms. Consequently, the calculations of the distribution of real wave height should be carried out for shorter periods, i.e. for seasonal or monthly averaged data on significant wave heights.

Figure 3 shows the field of wave height above mean level with a probability of 10

An increase in wave heights over March–May can be seen (Fig. 4) in the
Southern Hemisphere. Actually all the area of mid-latitudes from the
latitude of 40

Average annual significant wave height (m).

Wave height above mean level (m) with a cumulative probability of 10

Summer months (Fig. 5) are characterized by a general decrease in extreme wave probability. It is especially noticeable in the northern areas of the Atlantic and Pacific oceans. Also, wave heights slightly decreased in the Southern Hemisphere. It should be noted that storm tracks appear off the eastern coast of North America and disappear in the southern part of the Pacific Ocean. In addition, quite distinct trajectories of storms appear in the eastern part of the Pacific Ocean. Small wave heights can be observed in the Arctic Ocean, in the area free from ice.

During autumn months (Fig. 6) an increase in wave heights is observed in the Arctic Ocean, with values of the extreme wave height above mean level sometimes reaching 20 m. Among other features is an increase in the wave-free area in polar latitudes of the Southern Hemisphere, which is obviously connected with seasonal ice formation.

It is quite evident that the average monthly fields of cumulative wave height probability will allow us to obtain more exact information on the areas of extreme wave probability.

This paper describes a method of calculation of extreme wave probability,
based on (i) wave model runs for its relation to significant wave height
(Chalikov and Bulgakov, 2017) and (ii)

Wave height above mean level (m) with a cumulative probability of 10

Wave heights above mean level (m) with a cumulative probability of 10

The method uses the results of massive numerical simulations with 3-D irrotational wave models (Chalikov et al., 2014). Initial conditions for each run were assigned by the JONSWAP spectrum, but for each run random phases were different. Such details of the initial spectrum are not too important. The ensemble modelling is used to eliminate the effects of reversible non-linear interactions causing down shifting that can influence the statistics. To be sure that the simulated process can be treated as quasi-stationary; the time of integration was chosen to be relatively short, viz. 350 units of non-dimensional time. The extensive statistics were obtained by multiple repetitions of runs with the same initial spectrum. The total number of records used for construction of approximation (Eq. 3) was 4 587 520 000.

The wave spectrum during integration undergoes fluctuations: amplitudes grow with an increase in wave number due to reversible non-linear interactions. However, the averaged spectrum remains similar in different runs and more or less close to the spectrum assigned in initial conditions, confirming quasi-stationarity and some universality of the approximation (Eq. 3) to wave height probability. This approximation fills the gap between more or less known statistics on significant wave height and unknown statistics of real waves.

Wave heights above mean level (m) with a cumulative probability of 10

This method can be used for estimation of probability of extreme waves, which
is important for designing engineering constructions. The approach here
can be used to evaluate the height of waves of any given cumulative
probability. It is not expedient to use values less than 10

The maps of the global distribution of wave heights with a probability of 10

We do not state that results of this paper completely solve the problem of treating data on significant wave height in terms of real wave height (above mean level). The most difficult unresolved problem is the problem of estimating confidence intervals, which needs further extensive simulation and analysis.

Data are available at

KB conceived the main idea of the article, performed calculation of fields of wave height, and wrote the article. VK and DS carried out visualization of fields.

The authors declare that they have no conflict of interest.

The authors are thankful to Dmitry Chalikov for his useful consultations.

The investigation was fulfilled with financial support of the Russian Science Foundation (project no. 16-17-00124). Edited by: John M. Huthnance Reviewed by: two anonymous referees