Distribution of sea surface elevations in the form of a two-component Gaussian mixture

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Introduction

Sea surface waves are a weakly nonlinear process, and the statistical distributions of sea surface elevations and slopes are close to the Gaussian distribution [1]. Although deviations from the Gaussian distribution are small, they play an important role in applications related to ocean remote sensing [2, 3], as well as when forecasting the occurrence of anomalous waves [4].

As a rule, distributions based on truncated Gram–Charlier or Edgeworth series are used for the sea surface statistical description [5, 6]. The distributions are the expansion of the desired probability density function in Chebyshev–Hermite orthogonal polynomials. The use of truncated series leads to distortions in the desired probability density function due to the appearance of negative values in it, as well as several local maxima [7–9].

The relevance of the search for new approaches to the statistical description of the sea surface is determined by the fact that existing models do not make it possible to construct a probability density function of sea surface elevations over the entire range of their changes. One possible solution to this problem is to approximate the distribution of a quasi-Gaussian process by a two-component Gaussian mixture. Distributions of this type have not yet found wide application in oceanology, which may be due to the complex procedure for calculating their parameters [10]. For the first time, the use of such a model to describe the sea surface was independently proposed in [11, 12], in which probability density functions were constructed for the sea surface slopes. Recently, a two-component Gaussian mixture has been proposed to describe the distributions of sea surface elevations [13]. Unknown parameters for the desired Gaussian mixture are calculated based on the known statistical moments as in the construction of the Gram–Charlier and Edgeworth distributions.

This work aims at analyzing the possibility and limits of a two-component Gaussian mixture in order to describe the distribution of sea surface elevations. The analysis is based on direct measurements of sea waves carried out in the Black Sea.

Two-component Gaussian mixture

Finite Gaussian mixtures are widely used in various fields to approximate unknown probability density functions [9, 14]. The two-component Gaussian mixture of random variable ξ is as follows [15]

$P_S\left(\text{ξ}\right)=\sum _i\frac{{\text{α}}_i}{\sqrt{2\text{π}}{\text{σ}}_i}\text{exp}\left(-\frac{{\left(\text{ξ}-m_i\right)}^2}{2{{\text{\hspace{0.17em}σ}}_i}^2}\right),$   (1)

where α_i – weight of the i-th component (i = 1, 2), α_i ∈ (1, 2); m_i – expected value; σ_i² – variance. Weighting coefficients satisfy the condition

α₁ + α₂ = 1. (2)

Taking into account condition (2), it is necessary to find five parameters:
m₁, m₂, σ₁, σ₂ and α₁ to construct P_S(ξ). In [13], it was proposed to calculate them based on the first five statistical moments of sea surface elevations. The disadvantage of this approach is that according to wave measurements under marine conditions, as a rule, statistical moments are determined only up to the fourth order inclusive [16–18]. Therefore, we will use the first four statistical moments to calculate the model parameters (m₁, m₂, σ₁, σ₂) leaving the fifth parameter (α₁) free [11]. Parameter α₁ will be varied to satisfy the condition of distribution unimodality.

The procedure for calculating model parameters (1) is described in [10]. It amounts to solving the system of equations

${\text{α}}_1{m}_1+(1-{\text{α}}_1)m_2={\text{μ}}_1$,   (3)

${\text{α}}_1\left({m_1}^2+{{\text{σ}}_1}^2\right)+(1-{\text{α}}_1)\left({m_2}^2+{{\text{σ}}_2}^2\right)={\text{μ}}_2$,   (4)

${\text{α}}_1\left({m_1}^3+3m_1{{\text{σ}}_1}^2\right)+(1-{\text{α}}_1)\left({m_2}^3+3m_2{{\text{σ}}_2}^2\right)={\text{μ}}_3$,   (5)

${\text{α}}_1\left({m_1}^4+6{m_1}^2{{\text{σ}}_1}^2+3{{\text{σ}}_1}^4\right)+(1-{\text{α}}_1)\left({m_2}^4+6{m_2}^2{{\text{σ}}_2}^2+3{{\text{σ}}_2}^4\right)={\text{μ}}_4$,   (6)

where μ_i – statistical moment of order i :

${\text{μ}}_j=\int {\text{ξ}}^j{P}_S\left(\text{ξ}\right)d\text{ξ}$,

Let us assume that the average level of the surface is zero (μ₁ = 0), and the variance of the analyzed random variable is equal to 1 (μ₂ = 1). Parameters μ₃ and μ₄ – 3 are the skewness and excess kurtosis, respectively. System of equations (3)–(6) is symmetric with respect to triples of parameters (m₁, σ₁², α₁) and (m₂, σ₂², α₂).

Verification

To verify the model probability density function of sea surface elevations (1), the data of wave measurements obtained on a stationary oceanographic platform of Marine Hydrophysical Institute of RAS were used [19]. The measurements were carried out during December 2018. The platform was located in the Black Sea 600 m from the coast at a depth of about 30 m. The waves were measured with a string wave recorder [20].

The measurements were carried out under wind conditions that varied from calm to wind speed of 25 m/s. Significant wave heights (the average height of 1/3 of the highest waves) varied from 0.23 m to 2.26 m, the maximum wave height reached 4.9 m. The wavelengths corresponding to the peak of the wave spectrum ranged from 10 to 120 m.

The verification took place as follows. Continuous wave measurements were divided into wave records lasting 20 min. The total volume of data for analysis was more than 2200 wave records. Each wave record was centered and normalized so that its variance was equal to one, then experimental probability density function P_E (ξ) was calculated for each wave record. Statistical moments μ₃ = ⟨ ξ³⟩ and μ₄ = ⟨ξ⁴⟩ were also determined so that to calculate the parameters of two-component Gaussian mixture P_S (ξ). Here and below, symbol ⟨ ⟩ means averaging.

According to wave measurements previously carried out in the Black Sea, the values of statistical moments μ₃ and μ₄ can be found mainly in the following ranges [19]

$-0.2<{\text{μ}}_3<0.3$ и $2.6<{\text{μ}}_4<3.4$.   (7)

The same ranges were determined from measurements in the North Sea [18]. As a rule, exceeding the specified ranges occurs in situations where abnormally high waves (rogue waves) are observed. [17]. In this work, we will limit ourselves to the analysis of situations when μ₃ and μ₄ satisfy condition (7).

The experimental probability density function is calculated based on the analysis of the histogram of sea surface elevations. Width of intervals Dξ was taken equal to 0.45. Function P_E (ξ) was obtained from the histogram by normalizing it to the total number of points in the wave record and to the width of the interval.

The verification procedure for a two-component Gaussian mixture model consists of comparing functions P_E (ξ) and P_S (ξ). The criterion for the correspondence of model (1) to wave measurement data is relative error

$\text{ε}\left(\text{ξ}\right)=\frac{P_S\left(\text{ξ}\right)-P_E\left(\text{ξ}\right)}{P_E\left(\text{ξ}\right)},$

for which average value ⟨ε(ξ)⟩ and standard deviation δ (ξ) = ⟨(ε(ξ) – ⟨ε(ξ)⟩)²⟩^0.5 are calculated.

To calculate Gaussian mixture P_S (ξ), the procedure described in [10] was chosen. Taking into account condition (2), system of equations (3)–(6) was reduced to one sixth degree polynomial equation in m₁

$\begin{array}{l}2{{\alpha }_1}^2({\alpha }_1-{{\alpha }_1}^2-1){m_1}^6-4{\mu }_3{\alpha }_1(2{\alpha }_1-1){({\alpha }_1-1)}^2{m_1}^3+\\ +3({\mu }_4-3){\alpha }_1{({\alpha }_1-1)}^3{m_1}^2+{{\mu }_3}^2{({\alpha }_1-1)}^4=\mathrm{0,}\end{array}$     (8)

the solutions of which for given values μ₃ and μ₄ were found numerically by Newton’s method by varying α₁ The coefficients included in equation (8) were analyzed in [10], where it was shown that, except for the rare case when μ₃ = 0 and μ₄ > 3, it could always be solved and the construction of a probability density function was possible. From several solutions obtained for various possible α₁, the one was chosen that corresponded to the physical condition of unimodality of the resulting distribution and the positivity of values σ₁² и σ₂², which were recalculated, like m₂, from value m₁ according to the method discussed in [3]. Values μ₃ and μ₄ calculated for model Gaussian mixture P_S (ξ) obtained as a result of solving equation (8) were compared with the values calculated from the wave record and used in original equations (3)–(6). The accuracy of agreement between values μ₃ and μ₄ calculated from the Gaussian mixture and from the wave record is achieved no worse than 10^–3.

Figure 1 shows functions ⟨ε(ξ)⟩ and δ( ξ). Here, N(ξ) is number of points from which statistical characteristics were calculated in given interval Dξ. Functions ⟨ε(ξ)⟩ and δ( ξ) are average over the ensemble of situations in which measurements were carried out, with μ₃ and μ₄ satisfying condition (7). Parameters μ₃ = 0 and μ₄ > 3 in 11 wave records led to their exclusion from consideration for the reason stated above.

Analysis of deviations of model function P_S (ξ) from experimental one P_E (ξ) given in Fig. 1 indicates their smallness in the vicinity of point ξ = 0 and increase with | ξ |. As for range | ξ | < 3, the parameters characterizing this deviation satisfy conditions

$\left|〈\text{ε}\left(\text{ξ}\right)〉\right|<0.05$, $\text{δ}\left(\text{ξ}\right)<\mathrm{0.3.}$

For further analysis, all data were divided into groups corresponding to four ranges of the third statistical moment: group 1 – –0.2 < μ₃ £ 0, group 2 – 0 < μ₃ £ 0.1, group 3 – 0.1 < μ₃ £ 0.2, group 4 – 0.2 < μ₃ £ 0.3. Figure 2 shows variables ⟨ε(ξ, μ₃)⟩ and δ_i (ξ,μ₃) calculated for each group. Here, index i taking values from one to four corresponds to the group number. Parameter N_i (ξ, μ₃) shows the number of points from which the values ⟨ε_i (ξ, μ₃)⟩ and δ_i (ξ, μ₃) were calculated. Average value of relative error ⟨ε_i (ξ, μ₃)⟩ depends significantly on the group for which it was calculated. At the same time, standard deviation δ_i (ξ, μ₃) is almost the same for all groups. The discrepancy between P_S (ξ) and P_E (ξ) depends on how much statistical moment μ₃ deviates from the zero value corresponding to the Gaussian distribution. The greatest discrepancies are observed for group 4.

Fig. 1. Relative error ε(ξ) (a) and standard deviation δ(ξ) (b) calculated for an ensemble of situations, the number of points N(ξ) from which statistical characteristics were calculated in a given interval Dξ (c)

 

Fig. 2. Variables ε(ξ) (a), δ(ξ) (b), N(ξ) (c) calculated for four ranges μ₃: –0.2 < μ₃ £ 0 (blue), 0 < μ₃ £ 0.1 (red), 0.1 < μ₃ £ 0.2 (brown), 0.2 < μ₃ £ 0.3 (green)

 

We use a similar approach to analyze the approximation of the probability density of sea surface elevations for different values of the fourth statistical moment. Let us divide the data into groups corresponding to four ranges μ₄: group 1 – 2.6 < μ₄ £ 2.8, group 2 – 2.8 < μ₄ £ 3.0, group 3 – 3.0 < μ₄ £ 3.2, group 4 – 3.2 < μ₄ £ 3.4. Figure 3 shows variables ⟨ε_i (ξ, μ₄)⟩ and δ_i (ξ, μ₄) calculated for the specified groups.

 

Fig. 3. Variables ε(ξ) (a), δ(ξ) (b), N(ξ) (c) calculated for four ranges μ₄: 2.6 < μ₄ £ 2.8 (blue), 2.8 < μ₄ £ 3.0 (red), 3.0 < μ₄ £ 3.2 (brown), 3.2 < μ₄ £ 3.4 (green)

 

Division into groups according to the range of changes in statistical moments μ₃ and μ₄ results in a significant change in the relative error in the approximation of the probability density of sea surface elevations. In range | ξ | < 2, values δ_i (ξ, μ₃) and δ_i (ξ, μ₄) are 2–3 times lower than values δ(ξ) calculated for the entire ensemble of situations. This makes it possible to describe the probability density function by the semi-empirical relationship

$P_{\xi }\left(\text{ξ}\right)=P_S\left(\text{ξ}\right)\left(1+〈{\text{ε}}_E\left(\text{ξ}\right)〉\right)$,

where ⟨ε_E (ξ)⟩ is average relative error calculated for corresponding ranges μ₃ and μ₄.

Conclusion

The approximation of the probability density function of sea surface elevations by a two-component Gaussian mixture was verified for the values of the third and fourth statistical moments which vary within –0.2 < μ₃ < 0.3 and 2.6 < μ₄ < 3.4 and are characteristic of the Black Sea coastal zone. The criterion for the correctness of the approximation is the deviation of the model probability density function from that one calculated from wave measurement data, which is characterized by relative error.

In range | ξ | < 3, values of average relative error ⟨ε(ξ)⟩ and its standard deviation δ(ξ) are small and satisfy condition |⟨ε(ξ)⟩|  < 0.05, δ(ξ) < 0.3. Approximation error ⟨ε(ξ)⟩ has a systematic component which depends on the deviations of the third and fourth statistical moments from the values corresponding to the Gaussian distribution. A semi-empirical relationship has been constructed to take this component into account. The elimination of the systematic component will reduce δ(ξ), and the approximation accuracy can accordingly be increased by 2–3 times.

About the authors

Aleksandr S. Zapevalov

Marine Hydrophysical Institute of RAS

Author for correspondence.
Email: sevzepter@mail.ru
ORCID iD: 0000-0001-9942-2796
Scopus Author ID: 7004433476
ResearcherId: V-7880-2017

Chief Research Associate, Dr.Sci. (Phys.-Math.)

Russian Federation, 2 Kapitanskaya St., Sevastopol, 299011

Aleksandr S. Knyazkov

Marine Hydrophysical Institute of RAS

Email: fizfak83@yandex.ru
ORCID iD: 0000-0003-1119-1757
SPIN-code: 4254-4825

Leading Engineer

Russian Federation, 2 Kapitanskaya St., Sevastopol, 299011

References

Supplementary files