Simulation of sea surface elevations by the Edgeworth distribution

. The possibility of reliable monitoring of processes occurring near the ocean-atmosphere boundary is determined by how fully and correctly the geometric structure of the sea surface is described. The paper analyzes the possibilities of approximating the probability density function (PDF) of sea surface elevations by truncated Edgeworth series. The possibility of increasing the truncation order by using a regression equation linking the third and fifth statistical moments is considered. It is shown that the Edgeworth distribution, as well as the Gram-Charlier distribution, allows us to correctly describe PDF only in a limited region the width of which approximately corresponds to 1.25 significant wave height.


Introduction
Currently, the basis of surrounding environment monitoring is remote sensing. An important component of this process is the monitoring of the World Ocean. When sounding from spacecraft, information about processes in the ocean and in the atmosphere is read from the sea surface, therefore, one of the key tasks of modern oceanography is the correct description of the structure and variability of sea waves.
Sea waves are a weakly non-Gaussian process [1,2]. To approximate the probability density function (PDF) of such processes, as a rule, Gram-Charlier or Edgeworth series are used [3]. The coefficients of these series are calculated from known statistical moments. For elevations and slopes of the sea surface, statistical moments are usually known no older than the fourth order, therefore, series with a low truncation order are used [4,5,6]. A low truncation order leads to distortions at the tails of the distribution, which manifest themselves in the form of negative PDF values or the appearance of additional local maxima [7,8].
In recent decades, the relevance of information on the PDF of elevations of the sea surface has increased due to the development of methods and means of sounding the ocean from spacecraft [9,10]. Deviation of the PDF of sea surface elevations from the Gaussian distribution leads to an error in determining its level [11,12].
In this paper, we analyze the possibility of expanding the range of elevations in which PDF correctly describes the sea surface, using a regression dependence that determines the relationship of the third and fifth statistical moments. The Edgeworth and Gram-Charlier distributions are compared.

Edgeworth series
In the Gram-Charlier and Edgeworth distribution, the PDF is found by decomposing into a series of derivatives standard normal density function (1) Derivatives are described by the equation where ( )  (4) Usually, in applied tasks, a truncated Gram-Charlier series (type A) is used in the form where n  is a statistical moment of order n. It is assumed that the variance of a random variable is equal to one ( 1 2 =  ). In this case, 3  is a skewness, 3 4 −  is an excess kurtosis.
In our work we will use the Edgeworth series. Its advantage over the Gram-Charlier series is that it is a true asymptotic decomposition with a residual term of the order of the first discarded term [7].
We write the Edgeworth distribution in the form The terms of series (6) are grouped in such a way that each group contains members of the same order [13]. Group 2 and older groups describe the deviation from the Gaussian distribution caused by the nonlinearity of sea waves. Equation (6) will be rewritten in a more convenient form for analysis x In further analysis, we will take into account only the first four groups of additive component.

Data for simulation
The PDF of sea surface elevations should be unimodal and non-negative. When modeling PDF using Chebyshev-Hermite polynomials, these requirements are well met in the region, with relatively small values of elevations. Distortions occur in the region of high ridges and deep trough.
According to measurements made in different areas of the World Ocean, the values and mainly lie within [14,15] 3 .
The values of statistical moments go beyond the specified limits when abnormal waves (freak waves) are present on the surface. In these situations, the values 4  exceed 4 [16,17], the values 3  may be lower than -0.3 [18]. Measurements carried out in the field and laboratory conditions showed that 3  and 4  are not correlated with each other [19,20].
There are practically no data on changes in the fifth statistical moment. Therefore, for further analysis, we will use regression, which connects Regression (10) is based on the data of direct wave measurements carried out on a stationary oceanographic platform [20]. The platform is installed in the coastal zone of the Black Sea near the southern coast of Crimea. The depth at the place where it is located is about 30 m (corresponds to the condition of deep water). Meteorological conditions in the vicinity of the platform are described in [21].

Negative PDF values
Let us introduce the notation, let n PG be distinguished groups of terms in curly brackets of equation (7). Here the index n takes the values 2, 3 and 4. Then equation (7) can be rewritten as Let's start the analysis with the situation when the values of the statistical moments 3  and 4  are limiting, according to conditions (8) and (9). The PDF in the form (11) and (5) calculated for this value of the statistical moments is shown in Figs. 1. It can be seen that both ( )  To determine the values of x at which negative PDF values appear, it is necessary to find the roots of the equation The roots of the equations are denoted respectively as E R and GC R . When constructing Fig. 3, the values 4  were fixed, the roots of the equations E R and GC R are functions 3  .
The discontinuities of the curves correspond to the situation when the equations under study have no roots in the region 5  x .
It should be noted that the calculations were also carried out at 4 4 =  , which corresponds to the situation when abnormal waves are observed on the sea surface [17]. In this situation, negative values of the Edgeworth and Gram-Charlier distributions do not appear in the region 5  x .

Unimodality
As well as the presence of negative values, the appearance of local minima indicates the incorrectness of the PDF approximation in this region. The PDF of the sea surface elevations should monotonically fall in the direction of higher and lower values x relative to the position of the maximum. Similarly to the procedure for determining the boundary of the occurrence of negative PDF values, in order to find the region where the function is unimodal, it is necessary to find the roots of the equation Given (6), the PDF derivative of the Edgeworth distribution is represented as The derivatives of the Edgeworth and Gram-Charlier distributions are shown in Fig. 4.  The calculations were carried out at the same values 3  and 4  for which the distributions presented in Fig. 1 were calculated.
Calculations have shown that the boundaries of the area where PDF ceases to be unimodal for and approximately coincide. The approximations considered allow in different situations to build a PDF only in a limited area These boundaries of the region coincide with the estimates of the possibility of applying the Gram-Charlier distribution obtained for the slopes of the sea surface [22].

Conclusion
The analysis showed that PDF approximations based on the Edgeworth and Gram-Charlier series allow us to describe the distribution of sea surface elevation only in a limited region (14). Negative PDF values or local extremes appear outside this region. Local extremes mean that PDF becomes multimodal. The distortion is caused by the low order of truncation of the Edgeworth and Gram-Charlier series. Thus, the approximations do not describe the distribution of elevations in the area of high ridges and deep trough of a wave. The use of a regression equation relating the third and fifth statistical moments does not allow us to significantly expand the region (14). To solve this problem, other methods must be used. In particular, the PDF approximation of sea surface elevations by a Gaussian mixture can be such a solution.