Physical Oceanography, Vol. 13, No. 1, 2003
ANALYSIS OF OBSERVATIONS AND METHODS FOR CALCULATING
HYDROPHYSICAL FIELDS IN THE OCEAN
ANALYTIC MODEL OF THE PROBABILITY DENSITY OF SLOPES
OF THE SEA SURFACE
A. S. Zapevalov and Yu. B. Ratner
We study probability density models of the sea-surface slopes in which the experimentally ob-
served deviations from the normal distribution are taken into account. It is shown that the ap-
proximation of the probability density of slopes by finitely many terms of the Gram–Charlier
series (containing the first four moments of the distribution) is true only within a bounded range
of wind velocities. For the components of slopes, we propose a simple analytic model of prob-
ability density whose parameters are determined by using the experimental estimates of their var-
iance, skewness, and peakedness. The comparison with the data of field measurements shows
that the proposed model fairly well describes the actual distributions of the components of
The analysis of sea-surface slopes performed by using aerial photographs of the sun glitter shows that their
distribution is not strictly normal [1, 2]. Thus, the skewness of the component of slopes oriented along the wind
depends on the wind velocity and the values of peakedness noticeably differ from the values corresponding to
the normal distribution (namely,
for slopes oriented along the wind and
for crosswind slopes). The data
obtained with the use of two-dimensional laser slope meters [3, 4] give close estimates for the statistical mo-
ments of the components of slopes. This confirms the conclusion that their distribution is quasinormal.
The deviations from the normal distribution observed for small values of slopes play an important role in
the problems connected with scattering of light by the sea surface and, first of all, in the case of laser sounding
into nadir. However, at present, the interpretation of the data of laser sounding is most often based on the repre-
sentation of the water–air boundary in the form of a moving random Gaussian surface .
In , it is proposed to approximate the probability density of slopes by using the representation according
to which deviations from the normal distribution are described with the use of the Gram–Charlier series. The co-
efficients of this series are computed by using empirical estimates for the moments of the distribution of slopes.
As a rule, the experimental data enable us to determine these moments up to the fourth order, inclusively. Thus,
in practice, the accuracy of approximation of the distribution of slopes is limited by the use of the first five terms
of the Gram–Charlier series.
The results obtained by using finitely many terms of the Gram
Charlier series are not always satisfactory
and, in particular, one may even get negative values of the probability density . The problem of applicability
of finitely many terms of the Gram
Charlier series to the description of the probability distribution of slopes re-
quires additional investigations. In the present work, we make an attempt to solve this problem.
Marine Hydrophysical Institute, Ukrainian Academy of Sciences, Sevastopol. Translated from Morskoi Gidrofizicheskii Zhurnal,
3–17, January–February, 2003. Original article submitted June 6 2001; revision submitted August 20, 2001.
0928-5105/03/1301–0001 $25.00 © 2003 Plenum Publishing Corporation 1