Statistical Description of the Sea Surface by Two-Component Gaussian Mixture

A. S. Zapevalov✉, A. S. Knyazkov

Marine Hydrophysical Institute of RAS, Sevastopol, Russian Federation

✉ e-mail: sevzepter@mail.ru

Abstract

Purpose. The aim of the study is to analyze the possibility of applying the two-component Gaussian mixture with unequal dispersions in order to approximate the probability density function (PDF) of the sea surface elevation.

Methods and Results. The Gaussian mixture is constructed in the form of a sum of the Gaussians with different weights. Construction of the two-component Gaussian mixture with the regard for the condition imposed on the weight coefficients requires presetting of five parameters. The first four statistical moments of the sea surface elevations are applied for their calculation. The fifth parameter is used to fulfill the condition of unimodal distribution. To assess the possibility of using the approximations in the form of the Gaussian mixture, they were compared with the approximation based on the Gram – Charlier distribution, which was previously tested with direct wave measurement data. It is shown that at positive values of the excess kurtosis, in the range of a random value variation with a unit dispersion |ξ| < 3, two types of approximations are close; whereas at negative values of the excess kurtosis, noticeable discrepancies are observed in the area |ξ| < 1 (here ξ is the surface elevation normalized to the RMS value). Besides, it is also demonstrated that at the zero skewness, the PDF approximation in the form of the Gaussian mixture can be obtained only at the negative excess kurtosis.

Conclusions. At present, the models based on the truncated Gram – Charlier series, are usually applied to approximate the PDF elevations and slopes of the sea surface. Their disadvantage consists in the limited range, in which the distribution of the simulated characteristic can be described. The Gaussian mixtures are free from this disadvantage. A procedure for calculating their parameters is developed. To clarify the conditions under which the Gaussian mixtures can be used, direct comparison with the wave measurement data is required.

Keywords

sea surface, probability density function, Gaussian mixture, Gram-Charlier distribution, skewness, kurtosis

Acknowledgements

The study was carried out within the framework of the state assignment on theme No. 0555-2021-0004.

Original russian text

Original Russian Text © A. S. Zapevalov, A. S. Knyazkov, 2022, published in MORSKOY GIDROFIZICHESKIY ZHURNAL, Vol. 38, Iss. 4, pp. 422-431 (2022)

For citation

Zapevalov, A.S. and Knyazkov, A.S., 2022. Statistical Description of the Sea Surface by Two-Component Gaussian Mixture. Physical Oceanography, 29(4), pp. 395-403. doi:10.22449/1573-160X-2022-4-395-403

DOI

10.22449/1573-160X-2022-4-395-403

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