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

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 elevations. 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 vertical bar xi vertical bar < 3 two types of approximations are close; whereas at negative values of the excess kurtosis, noticeable discrepancies are observed in the area vertical bar xi vertical bar <1 (here xi 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.

Automated summary

The aim of this study is to investigate the potential of using a two-component Gaussian mixture with unequal dispersions as an approximation for the probability density function (PDF) of sea surface elevations. The Gaussian mixture is constructed using different weights, and its parameters are calculated based on the statistical moments of the sea surface elevations. Comparisons with the Gram-Charlier distribution approximation show that the Gaussian mixture performs well under certain conditions. However, further comparisons with wave measurement data are needed to determine the conditions under which the Gaussian mixture can be used effectively.