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13Original Russian Text © A. S. Zapevalov, A. S. Knyazkov, 2022, published in MORSKOY GIDROFIZICHESKIY ZHURNAL, Vol. 38, Iss. 4 (2022)
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 H 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 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 < 3, two types of approximations are close; whereas at negative values of
the excess kurtosis, noticeable discrepancies are observed in the area < 1 (here £, 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.
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
© A. S. Zapevalov, A. S. Knyazkov, 2022
© Physical Oceanography, 2022
Introduction
Describing the probability density function (PDF) of elevations generated by sea surface waves, approximations based on the Gram-Charlier distribution [1] are most widely used. The fundamental problem of applying these approximations is related to the fact that in practice the Gram-Charlier distribution is used in a truncated form, which allows to describe the distribution only in a limited variation range of a random value [2]. The need to solve a wide range of applied problems, primarily related to remote sensing of the ocean from space [3-5], has ISSN 1573-160X PHYSICAL OCEANOGRAPHY VOL. 29 ISS. 4 (2022) 395
Q
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led to the search for new approaches to constructing PDF approximations of sea surface elevations.
For simulation of sea surface elevations, it was recently proposed to use Gaussian mixtures [6], which have long been widely used in other areas in fundamental and applied research [7-9]. Previously, Gaussian mixtures were applied to approximate the PDF of sea surface slopes [10, 11].
In the general case, the problems of determining the number of unimodality area modes and boundaries for a Gaussian mixture have not been solved [12, 13], so it is necessary to check the correctness of its use for each physical problem. The present paper is aimed to analyze the possibility of using and limitations for a two-component Gaussian mixture with different variances in the PDF approximation of sea surface elevations. The analysis is carried out for the ranges of variation of the third and fourth statistical moments of the sea surface elevation, determined from the Black Sea measurement data [14].
Two-component Gaussian mixture
A two-component Gaussian mixture of a random variable £ has the following form [13]:
pM a ( (£- m1 )2 V a2 { (£- m2 )21
PS (£ )=-rexp -V 2 eXP 9 2 , (1)
V2 no, 2 a, V2no2 ¿o2
1 V 1 / 2 V 2 /
where ai is the 7-th component weight (7 = 1, 2), ai e (0,1); m7 is the mean value; a,2 is the variance. The weight coefficients satisfy the condition
a1 + a 2 = 1. (2)
To construct model (1), taking into account condition (2), it is necessary to find five parameters: m1, m2, o12, o22, a1. In [6], it was proposed to calculate them from the first five statistical moments of sea surface elevations. The disadvantage of this approach is that statistical moments no older than the fourth order are determined from the data of field measurements. Therefore, following [11], the first four statistical moments will be used to calculate the model parameters, leaving the fifth parameter (a1) free. The parameter a1 will be varied to satisfy the unimodality condition in PDF.
The statistical moments of a random value £ are defined as
M, = j £ JP (£) d £.
For a two-component Gaussian mixture
M, = a1^7,1 + a2^7,2 , (3)
where m, , = j£ P{ (£)d £, p (£) are the first and second terms in the model (1).
A general system for calculating the parameters of Gaussian mixture was proposed in [15]. Further, it will be assumed that the analyzed random value
variation £ is equal to one. Taking the average surface level equal to zero, taking into account (2) and (3), a system of equations for calculating the parameters of the model (1) is obtained:
at m1+ (1 - a 1)m2 = 0, (4)
a 1 (m12 + a 12-1) + (1 - a1) (m22 + a22 -1) = 0 , (5)
a1 (m13 + 3m1 a12 - ^3) + (1 - a1) (m23 + 3m2 a22 - ) = 0 , (6)
a1 (m^ + 6m^ a12 + 3a14 - ) + (1- a1)(m24 + 6m22 a22 + 3a24- ) = 0 . (7)
The parameters ^3 and ^4 - 3 are the skewness and excess kurtosis, respectively. System (4) - (7) will be investigated for the values - 0.2 < ^3 < 0.4, -0.4 < ^ 4 - 3 < 0.4, which for the Black Sea corresponds to the ranges of their change for wind waves and swell [14].
Following the approach [16], we reduce the system (4) - (7) to one equation, successively excluding the unknowns. From the equation (4) we have m2 = a1 m1 / (a1 -1), then from the equations (5) - (7), introducing an intermediate unknown
( m,2 + a,2 -1) (m 22 + a 22 -1) p -1-= ^-2-(8)
we obtain
m13 + 3m1 a12 - ^ 3 m23 + 3m2 a 22 - ^ 3
4^22^4 4 ^ 2 2 4
m1 + 6m1 a1 + 3a1 - _ m2 + 6m2 a2 + 3a2 -
(9) (10)
m1 m2
Using the equation (8), we express the dispersions:
a12 = m1 p +1 - m12, a22 = m2 p+1 - m22.
Substituting the expressions for a2 and a2 in (9) and (10), after the transformations we obtain
m1m2(3p- 2(m1 + m2)) = -^3, mm2(3p2- 2(m12 + m1m2 + m22)) = -^4 + 3 .
After a symmetrical change of variables of w = m1 + m2 and v = m1m2 form we have
3pv-2vw = -3p2v-2v(w2 -v) = -+ 3 .
Combining these two equations, we obtain
6v3 - 2v2w2 - 4vw ^3 + 3(^4 - 3)v + ^32 = 0.
PHYSICAL OCEANOGRAPHY VOL. 29 ISS. 4 (2022) 397
By reverse substitution, we express variables w and v in terms of m1 and finally obtain
2 a12 (a1 - a12 -1)m16 - 4 m3a 1 (2 a1 - 1)(a1 -1)2 m13 +
+3(m 4 - 3)a1 (a 1 -1)3 m2 + m 32 (a 1 -1)4 = 0. (11)
Thus, the original system of equations (8) - (11) is reduced to one sixth degree equation with respect to variable m1 with known m3, m4 values and a free parameter a1.
Let us consider the general properties of the sparse polynomial (11) as its coefficients change. In general terms
b0 m16 + b3 m13 + b4 m12 + b6 = 0, (12)
where b0 = 2 a12 (a 1 - a 12-1); b3 = -4 m3a1 (2 a 1 - 1)(a 1 -1)2; b4 = 3(m4 - 3)a1 (a1 -1)3; b6 = M32 (a 1 -1)4. In the range a1 e (0,1), b0 < 0, b6 > 0, while b6 = 0 only when M3 = 0. In case when b6 > 0, since the polynomial degree is even, according to the Descartes rule, equation (12) has both positive and negative real roots, since there is a change of sign in the series of its coefficients. When b3 > 0 and b4 < 0, the number of sign changes is three, and in other cases - only one. The sign of the coefficient b4 depends only on the sign of m4 - 3, b3 can change sign when changing both m3 and a1.
Let us consider separately the case when m 3 = 0. Equation (12) takes the form of m12(b0m14 + b4) = 0 . Since b0 < 0, it has nonzero real solutions only for positive values of b4, which corresponds to the condition m4 - 3 < 0 . Consequently, when M3 = 0 and m 4 - 3 > 0 the two-component mixture model has no real solutions and cannot be used. This limitation of two-component mixtures is obtained in general form for any process with the indicated values m3 and m 4.
The values of m1, satisfying (11), are found numerically by Newton's method for given m3 and m4, varying a1. Some of the solutions obtained should be excluded based on the condition of o12 and o22positivity. For the obtained values of m1, satisfying (11), and corresponding to a1 using the original system (4) - (7) m2, o12 and o22 were calculated and Gaussian mixture PDF was built. In the general case, model (1) can be both unimodal and bimodal [13, 17]. Since the distribution of wind wave elevations is unimodal, the PDF derivative was additionally analyzed and only the parameter values were selected when PS (£) had a single extremum (this is equivalent to the unimodality condition).
Along with symmetry with respect to triples of numbers (m1, o12, a1) and (m2, o22, a2), the system of equations (8) - (10) has an additional symmetry property: replacing (m1, m2, m3) by (m1, m2, - m3 ) gives identical solutions, so it is
sufficient to analyze only for positive values of ^3, i.e. for positive values of the asymmetry coefficient. PDF approximations in the form (1) are shown in Fig. 1.
F i g. 1. PDF approximations by the Gaussian mixture
Comparison with Gram-Charlier distribution
Sea surface waves are a quasi-Gaussian process [1, 18, 19]. The probability density function of such a process with unit variance can be represented as follows [2]:
M 1 ( 1 ) Pg-c (£) = ZC Ht (£)-j= exp^^£2 J , (13)
where Ci are the series coefficients; Ht are the orthogonal Hermite /-order polynomials. C coefficients are calculated by statistical moments. Since the statistical moments of the sea surface elevations are known only up to the fourth order inclusive, instead of (13) we obtain
PG-C (£) = vkexp (-2 £2 ){l + 3 (£) + 4 (£)}. (14)
PDF approximations in the form (14) are shown in Fig. 2. Expansion of the function into a series that includes a relatively small number of terms leads to a narrowing of the region where this approximation is valid [2]. In particular, it can be seen that function PG_C (£) with ^3 and ^4 values determined in the field experiments can take negative values.
F i g. 2. PDF approximation by the Gram-Charlier distribution
Previously, the approximation (14) was compared with empirical PDFs of sea surface elevations obtained from measurements of sea waves carried out on a stationary oceanographic platform of Marine Hydrophysical Institute [20]. The relative error s averaged over the ensemble of situations for the range |£| < 3
lies in the range of - 0.02 ... 0.07. The scatter of s values in the domain |£| < 1
does not exceed the 0.08 level; outside the specified domain, the scatter grows rapidly.
Approximation (14) verified according to field measurements can be used for a preliminary assessment of the PS (£) correctness. The ratio
R(£) = PG-C (£)/PS (£) is shown in Fig. 3. It can be seen that in the case when
the excess kurtosis is less than zero, the functions PS (£) and PG-C (£) differ
noticeably. Moreover, differences are observed even in the domain |£| < 1 where, as
noted above, there was a coincidence of PG-C (£) with the data of wave
measurements. For positive values of excess kurtosis in the domain |£| < 3,
the functions PS (£) and PG-C (£) are close.
F i g. 3. Dependences of ratio R = Pa_c/PS on the parameters and p.4. Curves 1-5 correspond to the values from 0 to 0.4 with a step 0.1
The reliability evaluation of the PDF approximation of sea surface elevations by a Gaussian mixture comparing with a distribution constructed on the basis of a truncated Gram-Charlier distribution is preliminary. The next step should be a direct comparison of PS (£) with empirical PDFs of sea surface elevations.
Conclusion
The main results of the study carried out are as follows.
1. A technique for calculating the two-component Gaussian mixture parameters for approximation of PDF of sea surface elevations has been developed. The analysis was carried out for the ranges of changes in the third (^ 3) and fourth (^ 4) statistical moments of the sea surface elevations, determined from the data of direct wave measurements in the Black Sea.
2. Symmetry properties of the Gaussian mixture equations, which reduce the number of calculations, are distinguished. In general, it is shown that PDF PHYSICAL OCEANOGRAPHY VOL. 29 ISS. 4 (2022) 401
approximations in the form of a Gaussian mixture in a particular case ^ 3 = 0 can be obtained only under the condition ^4 < 3 .
3. A comparison between the PDF approximation in the form of a two-component Gaussian mixture and the approximation based on the truncated Gram-Charlier distributions is carried out. When ^4 > 3 in < 3 domain the approximations are close, at ^4 < 3 significant discrepancies are observed. To clarify the conditions under which Gaussian mixtures can be used, a direct comparison with wave measurement data is needed.
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About the authors:
Aleksandr S. Zapevalov, Chief Researcher, Marine Hydrophysical Institute of RAS (2 Kapitanskaya St., Sevastopol, Russian Federation, 299011), Dr. Sci. (Phis.-Math.), Scopus Author ID: 7004433476, Web of Science ResearcherID: V-7880-2017, sevzepter@mail.ru
Aleksandr S. Knyazkov, Leading Engineer, Marine Hydrophysical Institute of RAS (2 Kapitanskaya St., Sevastopol, Russian Federation, 299011), ORCID ID: 0000-0003-1119-1757, fizfak83@yandex.ru
contribution of the authors:
Aleksandr S. Zapevalov - formulation of the aims and tasks of the study; development of requirements imposed on the model; the paper text preparation; editing and supplementing the paper text
Aleksandr S. Knyazkov - obtaining a mathematical model, testing the model; supplementing to the paper text
The authors have read and approved the final manuscript.
The authors declare that they have no conflict of interest.
