Revisiting the Cox and Munk wave-slope statistics using IASI observations of the sea surface

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Revisiting the Cox and Munk wave-slope statistics

using IASI observations of the sea surface

Charles-Antoine Guérina, Virginie Capelleb, Jean-Michel Hartmannb

aMediterranean Institute of Oceanography (MIO), Université de Toulon, Aix-Marseille

Université, CNRS, IRD, Toulon, FRANCE.

bLaboratoire de Météorologie Dynamique/IPSL, CNRS, Ecole Polytechnique, Institut

polytechnique de Paris, Sorbonne université, PSL research university, F-91120

PALAISEAU, FRANCE.

Abstract

We use radiances collected from space by the Infrared Atmospheric Sounder

Interferometer (IASI) when looking down at ocean surfaces during the day

to remotely determine the probability distribution of wave slopes. This

is achieved by using about 300 channels between 3.6 and 4.0 µm and a

physically-based approach which properly takes the contribution of the re-

ﬂected solar radiation into account. Based on about 150 millions observa-

tions, the same number of wave-slope probabilities are retrieved for wind

speeds (at 10 m) up to 15 m/s. We revisit and discuss the methodology

proposed by Cox and Munk (CM) to derive their celebrated wave-slope prob-

ability distribution function (pdf) from photographs of the sun glitter. We

propose an original and robust approach for accurate retrievals of the 7 pa-

rameters appearing in the Gram-Charlier representation of the pdf. Our

results for the mean square slopes (MSSs) are fully compatible with those of

CM, and with the more recent results by Bréon and Henriot, but our lower

uncertainties enable to point out departures from the linear wind-speed de-

pendencies and a slight overestimation of the upwind MSS described by the

linear ﬁt of CM at moderate wind speed. Our skewness and kurtosis coeﬃ-

cients show clear inﬂuences of the wind speed, with a steady decrease of the

former and the alongwind kurtosis coeﬃcient being maximal at moderate

wind speeds, features that CM could not point out due to the limitations

of their measurements. We revisit the renormalization procedure employed

Email address: Correspondence: guerin@univ-tln.fr (Charles-Antoine Guérin)

Preprint submitted to Remote Sensing of the Environment September 20, 2023

arXiv:2210.05456v2 [physics.ao-ph] 19 Sep 2023

by CM to obtain the complete variances from truncated pdfs and show that

it imposes stringent conditions on the kurtosis coeﬃcients that allow to de-

termine them accurately, with wind-dependent values speciﬁc to the local

sea state. We also provide measurements of the shifted position of the most

probable slope as well as a demonstration of a qualitative change of regime

in the updown wind asymmetry of the wave-slope probability when the wind

speed increases.

1. Introduction

The wave-slope probability distribution function (pdf), which statistically

describes the tilts of the facets at the sea surface, has many implications in

geosciences. Indeed, since it determines the area of the water-air interface, it

must be known for calculations of the energy and mass exchanges between the

oceans and the atmosphere involved in meteorological and climate studies.

Knowing the pdf is also important for various remote sensing issues including

retrievals of the sea-surface temperature [e.g. (Merchant et al., 2009; Capelle

et al., 2022; Capelle and Hartmann, 2022)], of aerosols and clouds [e.g. (Sayer

et al., 2010; Ottaviani et al., 2013; Choi et al., 2019; Meyer and Platnick,

2010; Mieslinger et al., 2022)], of salinity [e.g. (Camps et al., 2006; Burrage

et al., 2010)], of oil spills and slicks [e.g. (Adamo et al., 2009; Myasoedov

et al., 2012; Pisano et al., 2015)], of the ocean color [e.g. (Wang and Bailey,

2001; Gilerson et al., 2018; Mikelsons et al., 2020)], and of the wind speed

[e.g. (Bréon and Henriot, 2006; Zhang et al., 2018; Nelson et al., 2020)]. In

addition to these examples, recall that the wave-slope statistics also enters

in the physical parametrization of spectral wave models through its second-

order moments [e.g. (Stopa et al., 2016)]. and that the key role that its plays

is proven by the very large number of papers citing the seminal study of Cox

and Munk (1954a) who proposed its ﬁrst accurate parameterization.

Due to its importance, the wave-slope pdf has been the subject of nu-

merous investigations. As far as ﬁeld observations of the sea surface are con-

cerned, the pioneer study was carried seventy years ago by Cox and Munk

(1954a,b) [see also (Cox and Munk, 1956) for a documented report on the

experiment], from now on denoted as CM, who used black-and-white pho-

tographs of the sun glitter taken from an aircraft. Despite the limits of the

technique used and the very limited number of observations (25 for clean sea),

this groundbreaking investigation enabled to parameterize the inﬂuences of

the wind speed on the up-, -down, and cross-wind slope probabilities, which

were found diﬀerent. The parameters of the proposed Gram-Charlier distri-

bution, i.e.: The mean square slopes, and the skewness and kurtosis (peaked-

ness) coeﬃcients, were described through linear (or constant) functions of the

wind speed (up to 14 m/s) at 12.5 meters above the surface. The main lim-

itations of this seminal work are that the measurements only provided the

relative variations of the intensity of the sun glitter, due to lack of absolute

calibration, and that no information was obtained for large wave-tilt angles

θwfor which the sun glint was drowned in the background radiations. The

maximum available value of θwwas between 15◦and 25◦depending on wind

speed. Extrapolations thus had to be made and the wave-slope pdf needed

to be normalized. Note that Wu (1972) later on reanalyzed the data of CM

and found that the inﬂuence of the wind on the isotropic distribution is, for

speeds below (resp. above) 7 m/s, less (resp. more) pronounced than origi-

nally found. A new parameterization was proposed, based on a two-branch

logarithmic function leading to probabilities diﬀering from the former ones by

several tens of % in some cases. Since then, several other ﬁeld investigations

have been carried out, which are presented below in chronological order. Note

that in all cases the information on the wind direction and speed was pro-

vided by collocated (in time and space) measurements. Hughes et al. (1977)

determined the wave-slope pdf from the refraction, measured by a camera

carried by a crane 10 m above the bow of a ship, of the beam emitted by

an underwater He-Ne laser. The results, obtained for 9 wind speeds between

3.6 and 8.2 m/s and wave tilts up to 22◦, are in substantial agreement with

those of CM. Then, Mermelstein et al. (1994) obtained slope probabilities,

for wind speeds up to 20 m/s, from the wave-height power spectral density

deduced from radar observations by Donelan and Pierson (1987). With re-

spect to that of CM, their pdf is signiﬁcantly broader in both the up- and

cross-wind directions, with less asymmetry for wind speeds above 5 m/s. A

few years later, by using their laser-glint measurements complemented by

those of Hwang and Shemdin (1988), Shaw and Churnside (1997) introduced

a correction to the pdf of CM in order to take the inﬂuence of the degree of

atmospheric stability near the sea surface into account. This was achieved by

multiplying the original mean square slopes by a function depending on the

Richardson number Ri. The results are identical to those of CM for Ri=0.15

but diﬀer by more than +40 % for neutral and unstable states (Ri≤0.),

while a -35 % correction applies for Ri≥0.27. Then, Ebuchi and Kizu (2002)

used 30 millions (0.55-0.75 µm) images from the Visible Infrared Spin-Scan

Radiometer (VISSR). The pdf obtained in the 0-10 m/s wind-speed range

is quite close to that of CM for the crosswind component, but very diﬀer-

ent (narrower) for the alongwind one, which leads to a much less pronounced

(and inverse) up- to cross-wind asymmetry. These diﬀerences were attributed

to the fact that the observations of CM were made under conditions of grow-

ing waves, a questionable explanation in view of the results of Bréon and

Henriot (2006) discussed below. Again, no probability was derived for steep

waves since the investigated tilt angles are all smaller than 30◦. Bréon and

Henriot (2006), from now on denoted as BH, carried an investigation using a

dataset of 6millions radiances at 0.865 µm collected by the Polarization and

Directionality of the Earth’s Reﬂectances (POLDER). This led to results,

for wind speeds up to 15 m/s, which conﬁrm those of CM, except for some

skewness coeﬃcients which were found to vary non-linearly with the wind

speed [also see the discussion in Munk (2009)], but suﬀer from similar limi-

tations with θw≲22◦and the use of extrapolations and of a normalization.

In Ross and Dion (2007), observations between 0.444 and 0.864 µm where

used to retrieve slope variances (but no analytical model was derived from

them) for the 0-14 m/s wind-speed range. The results follow the general

trend established by CM, but point out a non-linear trend with wind speed

for the crosswind direction, as well as, in the lower wind-speed regime, for

the upwind one. Finally, the most recent study was carried (Lenain et al.,

2019) by analysing the glint of an airborne laser. Information was retrieved

for wave-tilt angles up to 26◦in the 2-14 m/s wind-speed range, leading to

up- and cross-wind mean square slopes in good agreement with those of CM,

but to diﬀerences for the skewness and peakedness. Since observations in

the optical domain are much more sensitive to steep and small wave facets

that do probings at long wavelengths, no review on this last topic is made

here. Let us however mention that the wave-slope statistics estimated from

microwave radar observations, e.g. (Jackson et al., 1992; Vandemark et al.,

2004; Boisot et al., 2015; Nouguier et al., 2016; Chen et al., 2018), generally

fall in between those of CM for a clean and a slick sea surface, because the in-

formation on facets with dimensions smaller than the wavelength is lost. For

completeness, recall that, besides the above mentioned studies, other ones

are limited to tests of previously proposed pdfs, made using ﬁelds measure-

ments. Zhang and Wang (2010) compared sun-glint observations at 0.859,

1.24, and 2.13 µm from the Moderate Resolution Imaging Spectroradiome-

ter (MODIS) with many of the above mentioned parameterizations. They

concluded that the facet models of CM and BH lead to the best predictions.

The CM parameterization of the pdf was also tested in numerous ﬁeld in-

vestigations in a broad range of wavelengths, e.g. (Haimbach and Wu, 1985;

Su et al., 2002; Vandemark et al., 2004; Gatebe et al., 2005; Hauser et al.,

2008; Liang et al., 2010; Yurovskaya et al., 2013; Zhang et al., 2018). In most

cases, a globally satisfactory agreement was obtained provided that, for ob-

servations at wavelengths exceeding the dimensions of some wave facets, the

associated ﬁltering eﬀect is taken into account. In conclusion, the bibliogra-

phy analysis shows that there are large inconsistencies between some of the

proposed pdfs, as illustrated by Fig. 4 of Zhang and Wang (2010), where

the results obtained using the parameterizations of Mermelstein et al. (1994)

and Ebuchi and Kizu (2002) diﬀer by a factor of over 3, those from Cox and

Munk (1954a,b) and Bréon and Henriot (2006) being in-between. Although

the ﬁndings of Bréon and Henriot (2006), Zhang and Wang (2010), Ross and

Dion (2007), Zhang et al. (2018), and Liang et al. (2010) are in favor of the

of CM, some uncertainties remain on the latter. The major one results from

the use of extrapolations and of a renormalization. In addition, the issue

concerning the linear dependence of the mean square slopes on wind speed

remains open, some studies having suggested a diﬀerent behavior [e.g. (Ross

and Dion, 2007; Wu, 1972)]. Furthermore, the inﬂuence of the wind on the

kurtosis and skewness coeﬃcients remains very uncertain. These elements

indicate that further investigations are still needed and that using, for this

purpose, observations provided by wavelengths and an instrument diﬀerent

from those so far used is desirable.

In Capelle and Hartmann (2022), daytime spectra recorded by the In-

frared Atmospheric Sounder Interferometer (IASI) satellite instrument (Hilton

et al., 2012) were used to retrieve sea-surface temperatures (SSTs) from ra-

diances at various channels between 3.6 and 4.0 µm. This required to take

the contribution of solar photons reﬂected by the surface into account. This

was achieved, for each observation, by simultaneously determining the SST

and a parameter (directly related to the wave-slope probability) scaling the

computed solar contribution through ﬁts of the IASI data using a physically-

based forward model. Note that Capelle and Hartmann (2022) focused on

the SST and on the validation of the methodology proposed for its retrieval,

which imposed the use of a limited set of observations: Those for which a

collocated in-situ measurement of the temperature from a drifter is available.

The consequently relatively small number (105) of wave-slope probabilities

obtained forbid any reliable investigation of the pdf, but the comparisons

made with the parameterizations of CM and BH demonstrated a good agree-

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RevisitingtheCoxandMunkwave-slopestatisticsusingIASIobservationsoftheseasurfaceCharles-AntoineGuérina,VirginieCapelleb,Jean-MichelHartmannbaMediterraneanInstituteofOceanography(MIO),UniversitédeToulon,Aix-MarseilleUniversité,CNRS,IRD,Toulon,FRANCE.bLaboratoiredeMétéorologieDynamique/IPSL,CNRS,EcoleP...

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Revisiting the Cox and Munk wave-slope statistics using IASI observations of the sea surface

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