Statistical modeling of the sea surface in the presence of abnormal waves

. A wide range of fundamental and applied problems requires a detailed description of the statistics of abnormal sea waves (freak waves or rouge waves). These waves are characterized not only by a change in energy, but also by strong nonlinearity, leading to extreme values of the higher cumulants. The possibilities and limitations of modeling the probability density function (PDF) of sea surface elevations by a two-component Gaussian mixture at extreme values of skewness and excess kurtosis are analyzed. The parameters of a two-component Gaussian mixture are calculated from known values of statistical moments. Model PDFs in the form of a two-component Gaussian mixture are compared with PDFs based on direct wave measurement data, and also compared with the known Gram-Charlier distribution. It is shown that with positive values of the excess kurtosis, the PDF in the form of a two-component Gaussian mixture can be constructed at the limit values of the skewness and excess kurtosis obtained in different regions of the World Ocean. With large negative values of the kurtosis, the shape of the probability density function is strongly distorted, which indicates the limit of applicability of a two-component Gaussian mixture to the description of such situations. Keywords: Two-Component Gaussian Mixture, Sea Surface, Elevation Distribution.


Introduction
One of the most important characteristics of the sea surface is the elevation probability density function (PDF), which is used in solving a wide range of applied problems related to the impact of waves on ships, offshore platforms, and coastal structures.It is also used in tasks related to remote sensing of the ocean.Three main types of models are used for statistical description of sea waves.The first type is dynamic models developed on the basis of hydrodynamic equations [1][2][3].The second type is numerical spectral wave models [4,5].The third type is models developed within the framework of the theory of random processes [6][7][8][9].
Here we will limit ourselves to analyzing models of the third type.Unknown parameters of the third type of models are calculated based on known statistical moments.These models describe the sea surface well when deviations from the Gaussian distribution are sufficiently small [10].With large deviations, distortions of the PDF of surface elevations occur [11].
The relevance of constructing PDF elevations of the sea surface at large values of excess kurtosis is primarily related to the analysis of the probability of the appearance of abnormally high waves (freak waves or rouge waves).According to wave measurements carried out in different areas of the World Ocean, large values of kurtosis indicate a high probability of occurrence of freak waves [12][13][14].PDF studies with significant deviations of skewness from zero are of interest for remote sensing problems.These deviations lead to a displacement of the leading edge of the reflected radio pulse of the altimeter and an error in determining the level of the sea surface [15][16][17].This error caused by the deviation of the distribution of sea surface elevations from the Gauss distribution is called skewness bias.
Wave measurements in the coastal zone at relatively shallow depths, where the approximation of "deep water" is not performed, have shown the existence of freak waves whose trough is larger than the crest [18].According to wave measurements carried out in the coastal zone of the Baltic Sea, in situations where freak waves were recorded, waves were observed in 17.5%, in which the depression exceeded the crest by one and a half times [19].In the coastal zone of the Baltic Sea, in situations where freak waves were recorded, waves were observed in which the trough exceeded the crest by one and a half times [19].Measurements in the Western Black Sea region of Turkey (depth 12.5 m) showed for freak wave that skewness changing in the range −0.01 and −0.4 and the excess kurtosis in the range of 0.1-1.2[20].
The purpose of this work is to assess the possibility and limitations of using a twocomponent Gaussian mixture at extreme values of skewness and excess kurtosis of sea surface elevations.

Simulation and numerical analysis 2.1 Two-component Gaussian mixture
Gaussian mixtures are great features that are widely used for modeling in various fields of science and practice [9,21,22].PDF in the form of a two-component Gaussian mixture has the form [23]  The procedure for calculating the parameters of a function is based on a relation linking its statistical moments with the statistical moments of its components  are third and fourth statistical points.The statistical moment of order n is defined as Where the symbol ... means averaging.Here and elsewhere we will assume that is an excess kurtosis.The procedure for calculating the parameters of a two-component Gaussian mixture is described in detail in [10].

Gram-Charlier distribution
The approximation of PDF by Gram-Charlier series is based on the well-known decomposition into a series by derivatives of the function [24] Which are defined by the expression Where are orthogonal Chebyshev-Hermite polynomials of order n.In general, a PDF of a random variable with a unit variance can be represented in the form Where n C are the coefficients of the series.Coefficients n C are calculated based on statistical moments.For the elevation of the sea surface, as a rule, the statistical moments of the elevations of the sea surface are known only up to the fourth order inclusive, therefore, truncated series are used It should be emphasized that the use of truncated series leads to distortions at the tails of the distribution, which manifest themselves as negative PDF values or the appearance of local extremes [11,24].These distortions impose a limit on the range of acceptable values  in which the function can be constructed ) ( GC P .

Numerical Simulation
The functions calculated for several values of skewness and excess kurtosis are shown in Fig. 1.The values of statistical moments 3  equal to -0.2 and 0.3, as well as  The PDF of sea surface elevations must be non-negative and satisfy the requirement of single-modality.In a linear wave field, which is a superposition of sinusoidal waves, the distribution of surface elevations is described by the Gaussian distribution [27].Sea waves are a weakly nonlinear process, so deviations from the Gaussian distribution are relatively small.This imposes another requirement the PDF, the number of inflection points must be equal to two.
To control the single-modality and convexity-concavity of the PDF, the first and second derivatives of the function were also analyzed, since for some values of skewness and excess kurtosis, it is not possible to construct a single-modal distribution of a twocomponent Gaussian mixture [10] , although there is no inflection of the function, but its second derivative has additional local maxima (see Fig. 2).

Comparison with data
Let's compare the model and calculated from the measurement data of the probability density functions of sea surface elevations.To do this, we will use the data of direct wave measurements obtained on the stationary oceanographic platform of the Marine Hydrophysical Institute of the RAS [25].The data of continuous measurements were divided into sessions lasting 20 minutes, for each of which the values of skewness and excess kurtosis were calculated, according to which the model was built  

Comparison with the Gram-Charlier distribution
In the wave measurement data used here, the values of skewness and excess kurtosis are in the ranges 3 .
In some situations, in particular, when the approximation of deep water is not performed, when abnormal waves are present, the ranges of variation of the higher statistical moments are much wider [20,26].To test the function beyond the specified ranges, we will use a distribution based on a truncated Gram-Charlier series ) ( GC P [28].Taking into account the distortions on the tails of the distribution that occur when using truncated Gram-Charlier series [11,24], the comparison will be carried out in the region 5 . 2   .Previously, PDF in the form were verified on the data of direct wave measurements [26].It has been shown that the relative error is small in the region 5 . 2   . It should be noted that for the problems of the impact of waves on ships and coastal structures, large crest and large trough are of the greatest interest, which make the main contribution to the distribution tails.In altimetric determination of sea level from spacecraft, it is important to know the deviation of the median distribution of elevations relative to the level of the nonperturbed surface (relative to 0   ) [16,17].Thus, the comparison   According to [20] in shallow water (depth 12.5 m) in the presence of abnormal waves, the skewness can approach -0.4.In experiments on a stationary oceanographic platform (depth about 30 m), the skewness varies from -0.23 to 0.37 [26].
To control PDF distortion, Fig. 5 shows the second derivatives of the PDF.

Conclusion
Currently, the main model describing the statistical characteristics of sea surface elevations remains the model constructed on the basis of truncated Gram-Charlier series.The known limitations associated with its application stimulated the search for new approaches.One of such approaches is the use of the PDF of sea surface elevations in the form of a twocomponent Gaussian mixture.
The two-component Gaussian mixture describes well the elevations of the sea surface with relatively small values of skewness and excess kurtosis.With positive values of excess kurtosis, the PDF in the form of a two-component Gaussian mixture can be constructed at the limit values of skewness and excess kurtosis obtained in different regions of the World Ocean.With large negative values of the excess kurtosis, the shape of the probability density function is strongly distorted, which probably indicates the limit of applicability of the two-component Gaussian mixture model to the description of such situations.Perhaps we are at the boundary of compatibility of this model and the unimodality condition, which is a natural physical condition in the description of surface waves.
4 and 1 approximately correspond to their limit values observed in the coastal zone of the Black Sea[25,26].

Fig. 1 .
Fig. 1.PDF model in the form of a two-component Gaussian mixture.The red and blue curves are a two-component Gaussian mixture calculated at 3 .0  A and 2 .0   A ; green curve is Gaussian distribution.

Fig. 2 .
Fig. 2. The second derivative of PDF in the form of a two-component Gaussian mixture waveforms of each measurement session were normalized in such a way as to obtain a single variance.Next, a histogram of the distribution of elevations was constructed with a width 25 .0   .The function was obtained from the histogram by normalizing by the total number of points in the waveform and by the width   .
Conferences 376, 03021 (2023) https://doi.org/10.1051/e3sconf/202337603021ERSME-2023 Situations in which the kurtosis exceeded the level of 0.4 are relatively rare.During December 2018, when the wave data analyzed in our work were obtained, 15 similar situations were recorded with a total of 2232 measurement sessions.The maximum value of excess kurtosis was 0.80.The functions values of skewness and excess kurtosis are shown in Figs. 3.

Fig. 3 .
Fig. 3. Comparison of PDF in the form of a two-component Gaussian mixture (blue curve) and empirical PDF (red curve).
interest for the tasks of studying the sea by means of remote sensing.The approximations of the PDF constructed within the framework of two models for extreme values of skewness and excess kurtosis are shown in Fig.4.

Fig. 4 .
Fig. 4. Comparison of the functions    S P

Fig. 5 .
Fig. 5. Comparison of the second derivatives of the PDF in the form of a Gaussian mixture (solid curve) and the Gram-Charlier distribution (dashed lines).Red curves correspond 4 .0   A ; blue curves correspond 4 .0  A .