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

Charles-Antoine Guérin  Correspondence<EMAIL_ADDRESS>Virginie Capelle
Jean-Michel Hartmann

###### 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 $\displaystyle\mu$m
and a physically-based approach which properly takes the contribution of the
reflected solar radiation into account. Based on about 150 millions
observations, 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
probability distribution function (pdf) from photographs of the sun glitter.
We propose an original and robust approach for accurate retrievals of the 7
parameters 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
dependencies and a slight overestimation of the upwind MSS described by the
linear fit of CM at moderate wind speed. Our skewness and kurtosis
coefficients 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 by CM to
obtain the complete variances from truncated pdfs and show that it imposes
stringent conditions on the kurtosis coefficients that allow to determine 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.

††journal: Remote Sensing of the Environment

[a]organization=Mediterranean Institute of Oceanography (MIO), Université de
Toulon, Aix-Marseille Université, CNRS, IRD, addressline=Toulon,
country=FRANCE. [b]organization=Laboratoire de Météorologie Dynamique/IPSL,
CNRS, Ecole Polytechnique, Institut polytechnique de Paris, Sorbonne
université, PSL research university, addressline=F-91120 PALAISEAU,
country=FRANCE.

##  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 numerous
investigations. As far as field observations of the sea surface are concerned,
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 photographs 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 distribution,
i.e.: The mean square slopes, and the skewness and kurtosis (peakedness)
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
limitations 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 $\displaystyle\theta_{w}$ for which the sun glint was drowned in the
background radiations. The maximum available value of
$\displaystyle\theta_{w}$ was between $\displaystyle 15^{\circ}$ and
$\displaystyle 25^{\circ}$ 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 originally 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 provided 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 respect 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 $\displaystyle R_{i}$. The results are
identical to those of CM for $\displaystyle R_{i}$=0.15 but differ by more
than +40 % for neutral and unstable states ($\displaystyle R_{i}\leq 0.$),
while a -35 % correction applies for $\displaystyle R_{i}\geq 0.27$. Then,
Ebuchi and Kizu (2002) used $\displaystyle 30$ millions (0.55-0.75
$\displaystyle\mu$m) 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 different (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 growing 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
$\displaystyle 6$ millions radiances at 0.865 $\displaystyle\mu$m 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 limitations with $\displaystyle\theta_{w}\lesssim 22^{\circ}$ and the
use of extrapolations and of a normalization. In Ross and Dion (2007),
observations between 0.444 and 0.864 $\displaystyle\mu$m 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 $\displaystyle 26^{\circ}$ 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 information 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
measurements. Zhang and Wang (2010) compared sun-glint observations at 0.859,
1.24, and 2.13 $\displaystyle\mu$m from the Moderate Resolution Imaging
Spectroradiometer (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 investigations 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 observations at wavelengths exceeding the dimensions of some wave facets,
the associated filtering effect is taken into account. In conclusion, the
bibliography 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 Infrared
Atmospheric Sounder Interferometer (IASI) satellite instrument (Hilton et al.,
2012) were used to retrieve sea-surface temperatures (SSTs) from radiances at
various channels between 3.6 and 4.0 $\displaystyle\mu$m. 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 ($\displaystyle 10^{5}$) 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
agreement.

In the present study, now that the excellent agreement between the IASI-
obtained SSTs and in-situ measurements (Capelle and Hartmann, 2022) has given
us a high confidence in the robustness of the methodology used, we investigate
the wave-slope probabilities. This is made possible by the treatment of a much
larger set (about 150 million) of data, thanks to the release of the
collocation-with-drifters constraint. The remainder of this paper starts, in
Sec. 2, with a presentation the IASI spectra used. The forward model, its
input data, and the procedure applied to fit the measured radiances are then
described in Sec. 3, the main characteristics of the IASI-retrieved dataset of
wave-slope probabilities being the subject of Sec. 4. Section 5 then details
how the latter were analyzed, which is done after recalling the approach used
by CM. The results obtained are then presented and discussed in Sec. 6, where
comparisons with previous studies are made.

##  2 The IASI spectra

The satellite-based Infrared Atmospheric Sounding Interferometer (IASI)
collects spectra of the radiation coming from below (Hilton et al., 2012). It
provides extremely well-calibrated radiances at 8461 equally-spaced points
from 645 to 2760 cm-1 (15.5 to 3.6 $\displaystyle\mu$m), with a spectral
resolution of 0.50 cm-1 after apodization (for the Level-1c spectra used
here). The instantaneous field of view leads to a ground resolution of 12 km
at nadir over a swath of about 2200 km (+/-50 ∘). Three successive generations
of IASI fly onboard the Metop-A, -B, and -C series of polar-orbiting
satellites, with Level-1c data available since July 2007, February 2013, and
July 2019, respectively. All three satellites have a Local Time of descending
orbit node of 9:30 AM (+/-15 mn) at the Equator. The spectral and radiometric
stabilities of IASI are continuously controlled by the Centre National
d’Etudes Spatiales ensuring an absolute accuracy better than 0.5 K at 280 K on
brightness temperatures and a relative one lower than 2 ppm on wave numbers
(Blumstein et al., 2004, 2007) .

In the present study, a set of about 150 million IASI daytime observations was
used. The spectra, recorded from Metop-A (2008-2021) and -B (2015-2019) were
selected, as done previously (Capelle et al., 2022; Capelle and Hartmann,
2022), using the following criteria: (i) Only cloud- and aerosol-free data
were retained, selected by using criteria on brightness temperatures from IASI
and the collocated AVHRR instrument. (iii) We only kept observations close to
nadir ($\displaystyle\leq$ 30∘) in order to minimize potential analysis biases
resulting from the dependence (Masuda et al., 1988) of the sea-surface
emissivity on the wind speed. (iii) Only those spectra for which the value of
the wind speed at 10 m above the observed spot is smaller than 15 m/s were
retained for the analysis due to the increasing potential contamination by
whitecaps (Anguelova and Webster, 2006; Albert et al., 2016) and the reduced
number of observations at large wind speeds.

In many observations, the solar contribution to the measured radiances (and
the associated wave-slope probability) is small, except when the instrument
points close to the sun glint. This latter case corresponds, for the geometry
of Fig. 1 and with the convention that all zenith angles are positive, to
small values of both $\displaystyle|{\theta_{I}}-{\theta_{S}}|$ and
$\displaystyle|{\varphi_{I}}-{\varphi_{S}}$-180${}^{\circ}|$.

Figure 1: Observation geometry. The z axis is along the local vertical, x
points toward the East and y toward the North. The insert enlightens the fact
that, the refection of light on the surface being specular, only an ad-hoc
wave slope redirects the solar photons toward the instrument.

The daytime descending orbits being close to the North-South direction, the
swath roughly follows an East-West line. $\displaystyle\varphi_{I}$ is thus
close to 0∘ for a Westward pointing and to 180∘ for and Eastward one. All
observations being made at 9:30 AM Local Time (at the Equator), the sun is at
the East and a significant solar contribution can only be collected on this
side of the swath and when $\displaystyle\varphi_{S}$ is close to 0∘ while
$\displaystyle\theta_{S}$ is not to far outside the 0-30∘ range of retained
$\displaystyle\theta_{I}$ values. During the summer, large wave-slope
probabilities, which correspond to weakly tilted wave facets, will thus be
observed at moderate North latitudes only, because the sun inclination to the
North is too large (i.e. $\displaystyle{\varphi_{S}}$ is too far above 0∘) in
the Southern hemisphere, the situation being reversed in the winter. This is
confirmed by the retrieved wave-slope probability map displayed in Fig. 2.

Figure 2: Wave-slope probability (logarithmic scale) retrieved, as explained
in Sec. 3, from the spectra collected by IASI along the descending orbits of
16 July 2018. The red line on the central orbit indicates the center of the
swath (nadir pointing).

Using IASI radiances, which are all for wavelengths greater than 3.6
$\displaystyle\mu$m, for retrieving the wave-slope pdf from the information
brought by the reflected solar light may a priori seem a not optimal choice
when compared to observations made in the visible or near infrared [as done in
e.g. Cox and Munk (1954a, b); Bréon and Henriot (2006); Ebuchi and Kizu
(2002); Zhang and Wang (2010); Ross and Dion (2007)]. Indeed, at such
significantly shorter wavelengths, not only is the sun radiance larger, but
the thermal emissions of the ocean and atmosphere can be neglected. However,
using IASI data has several noticeable advantages. The first is that they are
extremely well calibrated, which enables to retrieve absolute values of the
pdf and thus to avoid the use of any normalization procedure. The second is
that the calibration is, for a given instrument, remarkably stable over time,
and that the consistency between the three generations of IASI onboard
Metop-A, -B, and -C is also extremely good. These characteristics,
demonstrated in Capelle et al. (2022), enable to use long time series and
several instruments, thus providing a very large dataset of wave-slope
probability values. Finally, the potential biases introduced by un-rejected
observations contaminated by aerosols or due to errors in the modeling of
light scattering in the atmosphere are reduced. Indeed, the involved cross-
sections are much smaller around 4 $\displaystyle\mu$m when compared with the
visible region. In the mid-infrared, only the aerosols composed of the largest
particles (such as mineral dust or sea-salt) can significantly contribute to
the collected radiances (by scattering or absorbing the photons), while
aerosols typical of pollution or biomass burning have a negligible impact
(Pierangelo et al., 2013). Mineral dust have the largest influence, but they
can be detected [e.g. (Capelle et al., 2018)] and the contaminated
observations can therefore be disregarded as done in the present study.
Concerning the sea-salt aerosols, they are principally located at the bottom
of the planetary boundary layer, a region of small impact on the radiances
collected in the spectral range considered here. Finally, the influence of the
eventual presence of white caps is also reduced, since their reflectivity is,
with respect to that of a clean sea surface, significantly less increased
around 4 $\displaystyle\mu$m (Salisbury et al., 1993; Salisbury and D’Aria,
1994) than it is (Frouin et al., 1996; Dierssen, 2019) at the short infrared
[used by Bréon and Henriot (2006)] and visible [used by Cox and Munk (1954a)]
wavelengths.

##  3 Slope probabilities retrieval procedure

Since details on the theoretical model and data used can be found a previous
study (Capelle and Hartmann, 2022) and are recalled in Appendix A, only the
main elements are given below.

###  3.1 Forward model

For the observing geometry schematized in Fig. 1, the theoretical expression
of the IASI radiance at wave number $\displaystyle\sigma$, which disregards
any light scattering in the atmosphere, is:

$\begin{split}I_{theo}(\sigma,\theta_{I},\varphi_{I},\theta_{S},\varphi_{S})=I_{surf}(\sigma,\theta_{I},\mathbf{T_{surf}})+I_{atm}^{+}(\sigma,\theta_{I})\\\
+I_{atm}^{-}(\sigma,\theta_{I})+\mathbf{P_{w}}(\theta_{I},\varphi_{I},\theta_{S},\varphi_{S})I_{sun}(\sigma,I,\theta_{I},\varphi_{I},\theta_{S},\varphi_{S}).\end{split}$
(1)

The first term , $\displaystyle I_{surf}$, is the contribution of the surface
emission which can be expressed in terms of the sea directional emissivity and
skin temperature (SST) ($\displaystyle T_{surf}$), and of the atmospheric
transmission from the surface to the instrument. The second term,
$\displaystyle I_{atm}^{+}$, which represents the emission of the atmospheric
column between the surface and the instrument, can be computed provided that
the vertical profiles of temperature, pressure and volume mixing ratios of
absorbing species are known. The third term, $\displaystyle I_{atm}^{-}$,
which results from the emission of the atmosphere toward the surface that is
reflected and transmitted up to the instrument, depends on the same variables,
complemented by the reflectivity of the surface. Finally, the last term
describes the contribution of reflected solar photons. As shown in Appendix A,
it involves the out-of-the-atmosphere sun radiance, atmospheric transmission,
bi-directional reflectivity of the sea surface, and probability $\displaystyle
P_{w}(...)$ for waves to have a slope enabling solar photons coming along the
$\displaystyle(\theta_{S},\varphi_{S})$ direction to be redirected toward the
instrument, along $\displaystyle(\theta_{I},\varphi_{I})$, after a specular
reflection at the air-water interface (see insert in Fig. 1). As highlighted
by the use of bold characters in Eq. (1), $\displaystyle T_{surf}$ and
$\displaystyle P_{w}$ are the only unknown quantities of the model, that will
be retrieved (see Sec. 3.3) from fits of IASI-measured spectra. Indeed, all
the other parameters needed for the calculation of the radiances (atmospheric
state, sea-surface emissivity and reflectivity, absorption coefficients of
absorbing gases, solar radiance) can be obtained from independent sources, as
explained below.

###  3.2 Input data used

The index of refraction for pure water of Downing and Williams (1975) was
used, after a shifting of +4 cm-1 and an increase of the real part by +6 10-3
(Friedman, 1969) in order to take salinity effects into account, to compute
the sea surface emissivity as done by Masuda et al. (1988), and the bi-
directional reflectivity as explained in Appendix A. In the 3.6 to 4.0
$\displaystyle\mu$m region used for the retrievals, the extraterrestrial solar
radiance was represented by a Planck function with a temperature of 5657 K,
consistent with Platnick and Fontenla (2008). The vertical profiles of
temperature, pressure, and volume mixing ratios of the absorbing species of
each atmosphere observed by IASI were obtained (Capelle and Hartmann, 2022)
using a proximity recognition within the Thermodynamic Initial Guess Retrieval
(TIGR) database (Chédin et al., 1985; Chevallier et al., 1998) for a selection
of 8 properly chosen channels. The spectroscopic line parameters of the
absorbing gases, needed for the computation of the atmospheric spectra, were
taken from the GEISA database (Jacquinet-Husson et al., 2016) [aeris]. They
were complemented by the MT_CKD (version 3.1) water-vapor continua (Mlawer et
al., 2012) [MTCKD], and the collision-induced absorption by N2 and the CO2
line-wings corrections were computed as in Hartmann et al. (2018). All these
data were used as inputs to the 4AOP radiative-transfer code (Scott and
Chédin, 1981; Chéruy et al., 1995) [https://4aop.aeris-data.fr/] for
predictions of the atmospheric emissions and transmissions.

###  3.3 Retrieval procedure and its validation

For the treatment of the observations, the spectral points of locally minimum
absorption selected (Capelle and Hartmann, 2022) in two windows around 3.7 and
4.0 $\displaystyle\mu$m have been retained (107 points between 2480 and 2528
cm-1 and 185 points from 2594 to 2760 cm-1). The analysis then consists, for
each IASI-measured spectrum $\displaystyle I_{IASI}$, in its least-square fit
by the theoretical expression in Eq. (1). In other words, the quantity

$\begin{split}\Sigma_{\sigma_{k}}\left[I_{IASI}(\sigma_{k},\theta_{I},\varphi_{I},\theta_{S},\varphi_{S})-I_{theo}(\sigma_{k},\theta_{I},\varphi_{I},\theta_{S},\varphi_{S},T_{surf},P_{w})\right]^{2},\end{split}$
(2)

where the sums extends over the 292 retained channels
$\displaystyle\sigma_{k}$, is minimized by simultaneously floating the
$\displaystyle T_{surf}$ and $\displaystyle P_{w}$. This fit has little
ambiguity thanks to the small correlation between the adjusted parameters
brought by the use of a spectral range over which the ratio of the radiances
coming from the sea-surface emission and from the sun varies by a factor of
over 3.

Typical comparisons between measured and adjusted brightness-temperatures for
observations with a small and a large solar contribution are displayed in Fig.
3. Recall that the brightness temperature $\displaystyle BT(\sigma)$
associated with a radiance $\displaystyle I(\sigma)$ is such that the
blackbody (Planck) function at wave number $\displaystyle\sigma$ and
temperature $\displaystyle BT(\sigma)$ is equal to $\displaystyle I(\sigma)$.
As can be seen, the agreement is very good and the effect of the sun, which
can lead to a significant increase of the BTs with wave number, is nicely
taken into account, demonstrating the qualities of the forward model and fit.
As a result, the statistical uncertainty on the probability $\displaystyle
P_{w}$ provided by the adjustment is very small, below 1% for the most
probable slopes thanks to a significant solar contribution to the observed
radiances [we found that an error was made in the evaluation of the
uncertainty on the $\displaystyle A$ parameter made in Capelle and Hartmann
(2022), leading to strongly overestimated values of $\displaystyle\Delta A$.
This led us to wrongly attribute the observe scatter of the results to
uncertainties, while a large part of it was likely due to the influence of
scene parameters other than the wind, as discussed in Sec. 6.3]. Note that the
differences between the BTs in the dark and bright observations of Fig. 3
correspond, for the former, to an increase, due to the sun, of the radiance at
the shortest wavelength by a factor of over 5. Finally recall that the
robustness of the procedure described above was validated (Capelle and
Hartmann, 2022) by successfully comparing the retrieved SSTs with collocated
in-situ temperature measurements from drifters. Indeed, this showed, on
average and after correction of the cool-skin effect (Fairall et al., 1996),
an agreement within typically 0.01 K. In addition, the preliminary comparisons
(Capelle and Hartmann, 2022) between the fitted wave-slope probabilities and
values obtained using the parameterizations of Cox and Munk (1954a) and Bréon
and Henriot (2006) demonstrated a good consistency.

Figure 3: Observed (red symbols) and calculated+fitted (blue symbols)
brightness temperatures for observations including a large (bright case: upper
series of points) and very small (dark case: lower series of points) solar
contribution.

##  4 The IASI-retrieved wave-slope probabilities dataset

As demonstrated by Cox and Munk (1954a, b), the analysis of wave-slope
probabilities and the building-up of a pdf require knowledge of the wind speed
and direction. This information could a priori be provided by the Advanced
SCATterometer [ASCAT, see (Figa-Saldana et al., 2002) and ASCAT] which flies
together with IASI onboard the Metop platforms. However the problem is that
ASCAT only looks from 336 to 886 km away from nadir, for zenith angles in the
direction perpendicular to the orbit within the +/-[$\displaystyle
25^{\circ}$,$\displaystyle 53^{\circ}$] range. Since we only kept (see Sec. 2)
observations for $\displaystyle-30^{\circ}\leq\theta_{I}\leq+30^{\circ}$,
imposing a collocation with ASCAT limits the scanned $\displaystyle\theta_{I}$
range to +/-[$\displaystyle 25^{\circ}$,$\displaystyle 30^{\circ}$]. This
would significantly reduce the number of retrieved wave-slope probabilities,
thus affecting the reliability of the obtained pdf. We therefore resorted to
the wind information from the European Centre for Medium-range Weather Forcast
(ECMWF) ERA5 reanalysis dataset (Hersbach et al., 2020) which provides
(Hersbach et al., 2018) world-wide values (at 10 m above the surface) on
hourly and $\displaystyle 0.25^{\circ}\times 0.25^{\circ}$ spatial-grid bases.
This is not optimal since these data come from a model and not from
measurements. In order to quantify the possible errors, we used all
measurements from ASCAT over oceans for January, April, July and November
2019, and compared them with the corresponding (in space and time) ERA5 data.
This shows that the mean errors on the wind speed remain below 1 m/s with
typical RMS values after correction for the mean bias of about 1 m/s (for the
3-15 m/s range considered in this study). For the wind direction, the mean
difference between ASCAT and ERA5 is within $\displaystyle\pm 4^{\circ}$ (with
a change of sign when switching from the North to the South Hemisphere) with a
RMS (after correction for the bias) which decreases with wind speed, from
about $\displaystyle 20^{\circ}$ for 3 m/s down to $\displaystyle 6^{\circ}$
for 15 m/s. Note that these findings, which show that the errors are moderate,
are consistent with those of Belmonte Rivas and Stoffelen (2019) and Sandu et
al. (2020).

For the analysis of the IASI-retrieved probabilities, the relevant variables
are the wave slopes along and perpendicular to the wind direction. Following
many previous studies we retain for the x axis the upwind direction. This
corresponds to horizontally rotating the frame of Fig. 1 by
-$\displaystyle\varphi_{W}$, where $\displaystyle\varphi_{W}$ is the azimuth
angle between the upwind and East directions. Then, the up- and cross-wind
slopes, $\displaystyle s_{u}$ and $\displaystyle s_{c}$, are given by:

$s_{u}=dz/dx=-\frac{\sin(\theta_{S})\cos(\varphi_{S}-\varphi_{W})+\sin(\theta_{I})\cos(\varphi_{I}-\varphi_{W})}{\cos(\theta_{S})+\cos(\theta_{I})}$
(3)
$s_{c}=dz/dy=-\frac{\sin(\theta_{S})\sin(\varphi_{S}-\varphi_{W})+\sin(\theta_{I})\sin(\varphi_{I}-\varphi_{W})}{\cos(\theta_{S})+\cos(\theta_{I})}$
(4)

The treatment of all the selected IASI spectra provided about 150 million
wave-slope probabilities, with, for each of them, values (see above) of the
wind speed (between 0 and 15 m/s) and direction. Figure 4 presents the number
of data points versus the wind speed (with 0.5 m/s bins) and the wave-tilt
$\displaystyle(s^{2}_{c}+s^{2}_{u})^{1/2}=\mathrm{tg}(\theta_{w})$. As can be
seen, the largest numbers of observations are obtained for moderate winds (5-8
m/s), which are as well known the most frequent, and intermediate wave slopes
($\displaystyle 0.3-0.4$, corresponding to $\displaystyle\theta_{w}$ between
$\displaystyle 17^{\circ}$ and $\displaystyle 22^{\circ}$). The very small
tilts are only rarely observed due to the constraints on
$\displaystyle\theta_{S}$ and $\displaystyle\varphi_{S}$ imposed by the
position of the sun at 9:30 AM LT and those on $\displaystyle\theta_{I}$ and
$\displaystyle\varphi_{I}$ resulting from the IASI observation geometry.

Looking at the dependence of the IASI retrieved probabilities $\displaystyle
p(s_{u},s_{c},U)$ for given wind speeds $\displaystyle U$ shows that they are
slightly biased in the tail. Indeed, as exemplified by Fig. 5, the probability
tends toward a small but non-zero constant value for large slopes and does not
keep decreasing with increasing tilt, as it should do. The origin of this
bias, which can be considered as a background floor to be rejected, is at this
stage not completely established. We hypothetize several, possibly concurrent,
causes for it: a) An insufficient accuracy in the estimation of small
probabilities through the above described retrieval procedure; b) The
contamination of the tail of the distribution by rare but intense reflections
due to near-breaking waves; c) The contribution of whitecaps and foam patches;
d) Other parasitic contributions not taken into account in the model, such as
aerosols and thin clouds that have not been rejected by our filtering
procedure. Note that the bias has been found increasing with wind speed, which
supports the hypotheses b) and/or c) but does not rule out other reasons at
this stage. The very likely influence of breaking and whitecaps will be
further discussed in Section 6.3.

Figure 4: Number of observations classified by wind speeds and wave tilt
values (top panel) and by wind speed only (lower panel). Figure 5: Measured
probabilities (blue dots) of alongwind slopes in the principal section
($\displaystyle s_{c}=0$) for a wind speed of 10 m/s displayed with linear
(top panel) and logarithmic (lower panel) scales. The least-square fit with 3
Fourier azimuthal harmonics [Eq. (5) is shown by the red lines. The black
dotted lines is the same fit after correction for the bias.

In order to correct for this problem, we used the fact that integrating
$\displaystyle p(s_{u},s_{c},U)$ over $\displaystyle s_{u}$ and $\displaystyle
s_{c}$ should lead to unity, the idea being to subtract a constant
$\displaystyle b(U)$ to $\displaystyle p(s_{u},s_{c},U)$ in order to fulfill
this condition. However, since the integration requires to interpolate among
the measured values, a fit of the latter must first be made. For this purpose,
it is convenient to consider the probabilities in polar coordinates, that is
$\displaystyle p(s,\theta)$ with $\displaystyle s=(s_{u}^{2}+s_{c}^{2})^{1/2}$
and $\displaystyle\theta=\mathrm{atan}(s_{c}/s_{u})$ measured anticlockwise
with respect to the alongwind axis (thus $\displaystyle\theta=0/\pi$ for the
upwind/downwind directions while $\displaystyle\theta=\pm\pi/2$ for the
crosswind ones). Then, for each wind speed, $\displaystyle p(s,\theta)$ can be
expressed as an azimuthal series:

$p(s,\theta)=a_{0}(s)+a_{1}(s)\cos\theta+a_{2}(s)\cos(2\theta),$ (5)

with $\displaystyle s>0$ and $\displaystyle-\pi<\theta\leq\pi$, where we limit
the expansion to order 2 since this is sufficient to account for both the
upwind-crosswind and upwind-downwind asymmetries. The up- ($\displaystyle
s_{u}>0,s_{c}=0$), down- ($\displaystyle s_{u}<0,s_{c}=0$) and cross-wind
($\displaystyle s_{u}=0$) probabilities are then:

$\begin{split}p_{up}(s)&=a_{0}(s)+a_{1}(s)+a_{2}(s),\\\
p_{do}(s)&=a_{0}(s)-a_{1}(s)+a_{2}(s),\\\
p_{cr}(s)&=a_{0}(s)-a_{2}(s),\end{split}$ (6)

the omnidirectional (or isotropic component) value being $\displaystyle
p_{omni}(s)=a_{0}(s)$.

In order to determine the values of $\displaystyle a_{i=0,1,2}(s)$, we first
sorted the data by wind speed using a $\displaystyle 0.5$ m/s step and
centered bins of width $\displaystyle\pm 0.25$ m/s. Next, within each
interval, the $\displaystyle p(s,\theta)$s were sorted by ascending values of
$\displaystyle s$, irrespective of the value of $\displaystyle\theta$, and
binned in ranges $\displaystyle[s_{i}-\Delta_{i}s/2,s_{i}+\Delta_{i}s/2]$
containing a fixed number of points (typically, a few thousands), hence using
a variable width $\displaystyle\Delta_{i}s$. Then, for each $\displaystyle
s_{i}$, $\displaystyle a_{0}(s_{i})$, $\displaystyle a_{1}(s_{i})$ and
$\displaystyle a_{2}(s_{i})$ were obtained from a least-square fit of the
measured data using Eq. (5), leading to the typical results (red lines) shown
in Fig. 6. Finally, the original IASI pdf $\displaystyle p(s_{u},s_{c})$ for
each wind speed was corrected and replaced by
$\displaystyle\tilde{p}(s_{u},s_{c})=\max(p(s_{u},s_{c})-b,0)$, with
$\displaystyle b$ such that:

$\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\tilde{p}(s_{u},s_{c})ds_{u}ds_{c}=\int_{0}^{\infty}2\pi
s\ \tilde{a}_{0}(s)ds=1,$ (7)

where $\displaystyle\tilde{a}_{0}(s)=max(a_{0}(s)-b,0)$. This leads to the
black lines in the example of Fig. 5. As seen, the subtraction of the small
estimated bias restores the expected rapidly decreasing tail of the
distribution with very little impact on the central part. The retrieved values
of $\displaystyle b$ vary between 0.15 and 0.195 and are found wind-dependent,
as expected from the wind-dependent nature of the observed bias. In order to
evaluate the slope $\displaystyle s_{0}$ below which the pdf can be considered
as accurate, we used the (classical) criterion that the signal should be at
least twice larger than the noise threshold (3 dB SNR), that is $\displaystyle
a_{0}(s)=2b$. This leads to values from $\displaystyle s_{0}$=0.32 (18∘) at 3
m/s to $\displaystyle s_{0}$=0.52 (27.5∘) at 15 m/s, which are slightly larger
than the maximal ones obtained by Cox and Munk (1954a, b, 1956) which range
from about 0.27 (15∘) to 0.47 (25∘) depending on wind speed. Note, however,
that the range of available slopes probed by IASI extends far beyond these
values, with a significant number of wave tilts exceeding $\displaystyle 1$
($\displaystyle 45^{\circ}$), so that steeper slopes in the pdf can be used
after correction of the bias. Nevertheless, our statistical analysis will be
mainly based on the most reliable range of slopes, which is limited by the
aforementioned values.

Finally note that, for the analysis and results presented in Secs. 5.2 and
6.1, respectively, the raw IASI data were replaced (but still denoted as IASI
data), for each wind speed, by those for the up-, down-, and cross-wind
directions computed for each binned slope $\displaystyle s_{i}$ by using Eqs.
(6) and the values of $\displaystyle a_{0}(s_{i})=\tilde{a}_{0}(s_{i})$,
$\displaystyle a_{1}(s_{i})$, and $\displaystyle a_{2}(s_{i})$ determined as
described above. This corrects for the bias in the tail of the distribution
and filters the scatter, as shown by comparing the black line with the blue
dots in Fig. 5.

Figure 6: Azimuthal variations of the measured probabilities (blue dots)
together with their least-square fit (red solid lines) using Eq. (5) for a
wind speed of 10 m/s and a slope $\displaystyle s_{i}$=0.29 ($\displaystyle
16.3^{\circ}$).

##  5 Analysis of the probabilities and modeling the pdf

We below first recall the main elements of the study of Cox and Munk (1954a,
b, 1956). This enables to introduce the main statistical parameters and to
discuss some challenging aspects of their estimation. We then present our
analysis approach which uses several original techniques.

###  5.1 Recalling the Cox and Munk (CM) analysis

In their celebrated investigations (Cox and Munk, 1954a, b, 1956), CM derived
the wave-slope distribution from the analysis of sun-glitter aerial
photographs. To date, their relationships between the up- and cross-wind mean
squared slopes (MSSs) and wind speed still remain the reference and the most
frequently used.

Recall that the up- and cross-wind MSSs, $\displaystyle m_{u}$ and
$\displaystyle m_{c}$, are, for a given wind speed, defined as the second
moments of the pdf, i.e.:

$m_{u}=\int_{-\infty}^{+\infty}s_{u}^{2}p(s_{u},s_{c})ds_{u}ds_{c},\
m_{c}=\int_{-\infty}^{+\infty}s_{c}^{2}p(s_{u},s_{c})ds_{u}ds_{c},$ (8)

the total MSS being $\displaystyle m_{t}=m_{u}+m_{c}$. The main findings of CM
concerning $\displaystyle p(s_{u},s_{c})$ are that: a) It is close to a
bidimensional normal distribution, with a null mean slope and $\displaystyle
m_{u}>m_{c}$ ; b) It is symmetrical with respect to the crosswind direction
and negatively skewed with respect to the upwind one ; c) With respect to a
Gaussian distribution, it is slightly more peaky around the origin
($\displaystyle s_{u}=s_{c}=0$), and decreases more slowly at large slopes. d)
$\displaystyle m_{u}$, $\displaystyle m_{c}$ and $\displaystyle m_{t}$
increase quasi-linearly with wind speed. CM also proposed to describe the
deviations from Gaussian by an expansion in terms of a Gram-Charlier (GC)
series, i.e.:

$\begin{split}&p(s_{u},s_{c})=(2\pi\sqrt{m_{u}m_{c}})^{-1}e^{-\frac{1}{2}u^{2}}e^{-\frac{1}{2}c^{2}}\times\left(1-\frac{C_{12}u(c^{2}-1)}{2}\right.\\\
&-\frac{C_{30}(u^{3}-3u)}{6}+\frac{C_{40}(u^{4}-6u^{2}+3)}{24}\\\
&\left.+\frac{C_{22}(u^{2}-1)(c^{2}-1)}{4}+\frac{C_{04}(c^{4}-6c^{2}+3)}{24}\right),\end{split}$
(9)

where $\displaystyle u=s_{u}/\sqrt{m_{u}}$ and $\displaystyle
c=s_{c}/\sqrt{m_{c}}$. $\displaystyle C_{12}$ and $\displaystyle C_{30}$ are
the skewness coefficients along the wind direction, which account for the
asymmetry of the distribution, while the kurtosis coefficients $\displaystyle
C_{40}$, $\displaystyle C_{22}$ and $\displaystyle C_{04}$ enable to describe
the peakedness. This non-Gaussian attribute has been shown important for the
accurate characterization of the near-nadir microwave radar cross-section
(Chapron et al., 2000; Hauser et al., 2008; Bringer et al., 2012; Boisot et
al., 2015; Chen et al., 2018). Note that, for consistence with many studies in
radar and optical remote sensing, we chose to define the x-axis, which is the
reference of azimuth, as the upwind direction. We therefore use switched
notations with respect to those of CM (and BH), who chose the y axis to point
upwind. This has no consequence in the analysis but this implies that the
index $\displaystyle i$ in our $\displaystyle C_{ij}$ refers to the upwind
axis while $\displaystyle j$ refers to the crosswind one (hence our
$\displaystyle C_{ji}$ corresponds to the $\displaystyle C_{ij}$ of CM and
BH).

Since the CM data were limited to small and moderate wave tilts, only
“incomplete” MSSs could be evaluated due to truncated integrations in Eq. (8),
limited to some wind-dependent threshold value $\displaystyle s_{0}$. For more
accurate estimations, CM carried an extrapolation based on the fit, to their
measured data, of an analytical representation of the logarithm of the
probabilities using truncated Fourier series of the azimuthal angle
$\displaystyle\theta$ (with respect to the upwind direction) and power series
of the wave slope $\displaystyle s=(s_{u}^{2}+s_{c}^{2})^{1/2}$, i.e.:

$\begin{split}&\log
p(s,\theta)=\alpha_{0}-\alpha_{0}^{\prime}s^{2}+\alpha_{0}^{\prime\prime}s^{4}+s(\alpha_{1}+\alpha_{1}^{\prime}s^{2})\cos\theta\\\
&+s^{2}(\alpha_{2}+\alpha_{2}^{\prime}s^{2})\cos(2\theta),\end{split}$ (10)

A key parameter in this expansion is the coefficient
$\displaystyle\alpha_{0}^{\prime}$, corresponding to the quadratic term of the
omnidirectional (i.e, azimuthally averaged) pdf, which is an inverse MSS
($\displaystyle\alpha_{0}^{\prime}=1/m_{t}$ under an isotropic Gaussian
assumption). This coefficient, which can be estimated quite easily from a fit
of the measured data around the origin ($\displaystyle s=\theta=0$), drives
the decay of the main quasi-Gaussian lobe of the distribution and therefore
also controls the available range of values for the wave tilts which are
limited by the dynamical range of the pdf values above the noise threshold.
with CM experimental data extending to $\displaystyle s_{0}\simeq
s_{4}=(4/\alpha_{0}^{\prime})^{1/2}$. Assuming Eq. (10) to hold beyond the
limiting slope $\displaystyle s_{4}$ and determining the coefficients
$\displaystyle\alpha_{j}^{,^{\prime},^{\prime\prime}}$ by fitting the data in
azimuth ($\displaystyle\theta$) and wave slope ($\displaystyle s$) using Eq.
(10), CM were then able to extrapolate and obtain more reliable MSSs. They
observed the convergence of the integrals in Eq. (8) for $\displaystyle
s_{8}=(8/\alpha_{0}^{\prime})^{1/2}$ and found wind-independent factors with
respect to the incomplete values calculated at $\displaystyle s_{4}$, i.e.:

$\begin{split}&m_{u}\simeq 1.23\int_{|s|\leq
s_{4}}s_{u}^{2}p(s_{u},s_{c})ds_{u}ds_{c}\\\ &m_{c}\simeq 1.22\int_{|s|\leq
s_{4}}s_{c}^{2}p(s_{u},s_{c})ds_{u}ds_{c}.\end{split}$ (11)

Finally, the following famous formula were proposed for wind speeds at the
“mast heigth’“ of 12.5 m in the 1-14 m/s range (Cox and Munk, 1954a):

$\begin{split}m_{u}=3.16\ 10^{-3}\ U_{12.5}\quad,\quad m_{c}=3.\ 10^{-3}+1.92\
10^{-3}U_{12.5}.\end{split}$ (12)

The skewness ($\displaystyle C_{12}$, $\displaystyle C_{30}$) and kurtosis
($\displaystyle C_{40}$, $\displaystyle C_{04}$, $\displaystyle C_{22}$)
coefficients were deduced from the fitted coefficients in Eq. (10) and the
MSSs, but with a poor accuracy. Recall that the CM results were confirmed by
Bréon and Henriot (2006) using wave-slope probabilities, retrieved from
satellite observations, directly fitted with the GC model of Eq. (9) floating
7 parameters (the 2 directional MSSs, the 2 skewness and 3 kurtosis
coefficients). The values of $\displaystyle m_{u}$ and $\displaystyle m_{c}$
obtained are very close to those of CM while some GC coefficients show a
dependence on wind speed that CM could not establish due to insufficient
accuracy. Our attempts to carry similar fits of the IASI-retrieved
probabilities led to ambiguous solutions due to the numerous floated
parameters and the significant correlations between them. We found, in
particular, that variations in the estimated MSSs and kurtosis parameters can
compensate each other through the quadratic terms in Eq. (9) and can only be
separated by the quartic terms that enter in play for larger values of the
wave tilt, so that the optimization procedure is not sufficiently robust and
can lead to an ambiguous set of solutions. In order to overcome this problem,
We devised an adapted and original procedure for the evaluation of the MSSs
and GC coefficients, which is described below.

###  5.2 An original approach for the analysis of the slope probabilities

Retaining the GC expansion in Eq. (9) we here present a methodology for robust
estimations of the directional MSSs as well as of the skewness and kurtosis
coefficients. Our approach relies on the use of the azimuthal expansion [Eq.
(5)] to fit the measured distribution rather than its logarithm, as was done
by CM [Eq. (10]. This combines several advantages: a) Taking the logarithm of
the distribution can produce very large negative values that give an excessive
weight to small probabilities in a least-square fitting of the Fourier
harmonics; b) The moments of the azimuthal functions are useful for the
estimation of the directional MSSs and GC coefficients, as shown later; c)
This representation enables to separate the symmetric and anti-symmetric
components of the alongwind distribution:

$\begin{split}p_{sym}(s_{u},s_{c})&=\frac{p(s_{u},s_{c})+p(-s_{u},s_{c})}{2}=a_{0}(s)+a_{2}(s)\cos
2\theta,\\\
p_{asym}(s_{u},s_{c})&=\frac{p(s_{u},s_{c})-p(-s_{u},s_{c})}{2}=a_{1}(s)\cos\theta,\end{split}$
(13)

which facilitates the identification of the skewness and kurtosis coefficients
in the GC model as $\displaystyle p_{sym}$ (resp. $\displaystyle p_{asym}$) is
obtained by setting $\displaystyle C_{12}=C_{30}=0$ (resp. $\displaystyle
C_{40}=C_{04}=C_{22}=0$) in Eq. (9).

####  5.2.1 Estimation of the mean square slopes (MSSs)

As discussed above, the estimation of the MSSs from experimentally-determined
wave-slope probabilities is a challenging problem due to the uncertainties
affecting the measured pdf and the lack of reliable data beyond a given slope,
which led CM to use extrapolations and a renormalization, a procedure which
has been criticized by further analysis (Wentz, 1976). We propose a different
and original approach based on a direct calculation (and not a fit) of the
MSSs using an appropriate combination of moments $\displaystyle\cal
M\mit_{kn}$ of the azimuthal functions, defined by:

${\cal M\mit}_{kn}=\int_{0}^{\infty}s^{k}\ a_{n}(s)ds.$ (14)

Note that the normalization of the pdf implies:

$2\pi{\cal M\mit}_{10}=1,$ (15)

and that the directional MSSs are:

$\begin{split}m_{u}=\pi({\cal M\mit}_{30}+{\cal M\mit}_{32}/2)\quad,\quad
m_{c}=\pi({\cal M\mit}_{30}-{\cal M\mit}_{32}/2).\end{split}$ (16)

As discussed in Sec. 4, the IASI-retrieved probabilities are potentially
biased (and significantly contaminated by noise) for wave-tilt angles greater
than 18∘-34∘ depending on the wind speed. For these limit values, the
integrals defining the moments [Eq. (14)] are not always converged.
Furthermore, since the relative contribution of steep waves to
$\displaystyle{\cal M\mit}_{kn}$ increases with increasing $\displaystyle k$,
so does the bias on $\displaystyle{\cal M\mit}_{kn}$ due to truncation, and
using the third-order ones $\displaystyle{\cal M\mit}_{30}$ and
$\displaystyle{\cal M\mit}_{32}$ to determine the MSSs from Eq. (16) is thus
inappropriate. This led us to consider some lower moments of the distribution
also involving the MSSs. Indeed, one has:

$\begin{split}\int_{0}^{+\infty}p_{sym}(s_{u},0)ds_{u}&={\cal
M\mit}_{00}+{\cal M\mit}_{02}=\frac{1}{2}(2\pi
m_{c})^{-1/2}\left(1+\frac{C_{04}}{8}\right),\\\
\int_{0}^{+\infty}s_{u}p_{sym}(s_{u},0)ds_{u}&={\cal M\mit}_{10}+{\cal
M\mit}_{12}=\sqrt{\frac{m_{u}}{2\pi}}(2\pi m_{c})^{-1/2}\\\
&\times\left(1+\frac{1}{8}C_{04}-\frac{1}{24}C_{40}-\frac{1}{4}C_{22}\right),\\\
\int_{0}^{+\infty}p_{sym}(0,s_{c})ds_{c}&={\cal M\mit}_{00}-{\cal
M\mit}_{02}=\frac{1}{2}(2\pi m_{u})^{-1/2}\left(1+\frac{C_{40}}{8}\right),\\\
\int_{0}^{+\infty}s_{c}p_{sym}(0,s_{c})ds_{c}&={\cal M\mit}_{10}-{\cal
M\mit}_{12}=\sqrt{\frac{m_{c}}{2\pi}}(2\pi m_{u})^{-1/2}\\\
&\times\left(1+\frac{1}{8}C_{40}-\frac{1}{24}C_{04}-\frac{1}{4}C_{22}\right),\\\
\end{split}$ (17)

from which one obtains:

$\begin{split}m_{u}=\frac{\pi}{2}F_{u}\times\left(\frac{{\cal
M\mit}_{10}+{\cal M\mit}_{12}}{{\cal M\mit}_{00}+{\cal
M\mit}_{02}}\right)^{2}\quad,\quad
m_{c}=\frac{\pi}{2}F_{c}\times\left(\frac{{\cal M\mit}_{10}-{\cal
M\mit}_{12}}{{\cal M\mit}_{00}-{\cal M\mit}_{02}}\right)^{2},\end{split}$ (18)

with

$\begin{split}&F_{u}=\left(\frac{1+\frac{1}{8}C_{04}}{1+\frac{1}{8}C_{04}-\frac{1}{24}C_{40}-\frac{1}{4}C_{22}}\right)^{2},\\\
&F_{c}=\left(\frac{1+\frac{1}{8}C_{40}}{1+\frac{1}{8}C_{40}-\frac{1}{24}C_{04}-\frac{1}{4}C_{22}}\right)^{2}.\end{split}$
(19)

Anticipating that $\displaystyle F_{u}$ and $\displaystyle F_{c}$ have values
very close to $\displaystyle 1$ (as confirmed later) we obtain, using Eqs.
(15) and (18):

$\begin{split}m_{u}\simeq m_{u}^{0}=\frac{1}{8\pi}\left(\frac{1+2\pi{\cal
M\mit}_{12}}{{\cal M\mit}_{00}+{\cal M\mit}_{02}}\right)^{2}\quad,\quad
m_{c}\simeq m_{c}^{0}=\frac{1}{8\pi}\left(\frac{1-2\pi{\cal M\mit}_{12}}{{\cal
M\mit}_{00}-{\cal M\mit}_{02}}\right)^{2}.\end{split}$ (20)

These equations involve moments of order $\displaystyle k=0$ and
$\displaystyle 1$, which converge rapidly and thus provide good first guesses
($\displaystyle m_{u}^{0}$ and $\displaystyle m_{c}^{0}$) of $\displaystyle
m_{u}$ and $\displaystyle m_{c}$ (which will be refined after estimation of
the factors $\displaystyle F_{u}$ and $\displaystyle F_{c}$). A careful
investigation of the cumulative sums involved in the moments shows that
$\displaystyle m_{u}^{0}$ and $\displaystyle m_{c}^{0}$ in Eq. (20) are indeed
correctly evaluated at the limiting slope of the IASI data, which is not the
case when Eq. (16) is used.

####  5.2.2 Estimation of the kurtosis coefficients

In order to determine $\displaystyle F_{u}$ and $\displaystyle F_{c}$ from Eq.
(19), we must estimate $\displaystyle C_{40}$, $\displaystyle C_{04}$, and
$\displaystyle C_{22}$ which appear in Eq. (9). Those are actually the excess
kurtosis of the two-dimensional distribution, given by:

$\begin{split}C_{40}&=m_{u}^{-2}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}s_{u}^{4}p(s_{u},s_{c})ds_{u}ds_{c}-3,\\\
C_{04}&=m_{c}^{-2}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}s_{c}^{4}p(s_{u},s_{c})ds_{u}ds_{c}-3,\\\
C_{22}&=(m_{u}m_{c})^{-1}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}s_{u}^{2}s_{c}^{2}p(s_{u},s_{c})ds_{u}ds_{c}-1,\end{split}$
(21)

which involve moments, of orders 4 and 5 in Cartesian and polar coordinates,
respectively, which cannot be reliably calculated due to the uncertainties
affecting the tail of the distribution. Nevertheless, $\displaystyle C_{40}$,
$\displaystyle C_{04}$, and $\displaystyle C_{22}$ can also be obtained from
direct fits of the IASI probabilities using the GC model, provided that the
MSSs are known and can be fixed. This was done using the values of
$\displaystyle m_{u}^{0}$ and $\displaystyle m_{c}^{0}$ obtained from Eq. (20)
as first guesses, before refining the estimations. We thus determined
$\displaystyle C_{40}$, $\displaystyle C_{04}$, and $\displaystyle C_{22}$
through least-square adjustments of the measured symmetrical components
$\displaystyle p_{sym}$ along the 3 principal directions
($\displaystyle\theta=0^{\circ},\ 90^{\circ},\ 45^{\circ}$) using Eq. (13) and
the associated expressions of the GC model. Then, the final MSSs were obtained
from:

$\begin{split}m_{u}=F_{u}(m_{u}^{0},m_{c}^{0})\times m_{u}^{0}\qquad,\qquad
m_{c}=F_{c}(m_{u}^{0},m_{c}^{0})\times m_{c}^{0},\end{split}$ (22)

based on factors $\displaystyle F_{u}$ and $\displaystyle F_{c}$ computed
using Eq. (19) and the obtained values of $\displaystyle
C_{40}(m_{u}^{0},m_{c}^{0})$, $\displaystyle C_{04}(m_{u}^{0},m_{c}^{0})$ and
$\displaystyle C_{22}(m_{u}^{0},m_{c}^{0})$. Finally, refitting the kurtosis
coefficients on the basis of the refined MSSs yielded the definitive best
estimates $\displaystyle C_{40}(m_{u},m_{c})$, $\displaystyle
C_{04}(m_{u},m_{c})$ and $\displaystyle C_{22}(m_{u},m_{c})$. Based on our
estimated kurtosis coefficients (see further), we found the corrective factors
$\displaystyle F_{u}$ and $\displaystyle F_{c}$ very close to $\displaystyle
1$ (within $\displaystyle 3\%$) so that the approximated MSSs of Eq. (20) are
very close to the refined ones in Eq. (18).

####  5.2.3 Estimation of the skewness coefficients

The skewness of the distribution is accounted for by the $\displaystyle a_{1}$
function in Eq. (5). Similarly to the kurtosis coefficients, the skewness
parameters $\displaystyle C_{30}$ and $\displaystyle C_{12}$ can be obtained
by fitting the asymmetric component $\displaystyle p_{asym}$ of the measured
distribution derived from Eq. (13) with that of the GC model once the MSSs are
known. However, they can also be derived from a direct calculation using the
moment $\displaystyle{\cal M\mit}_{11}$, since one has:

$\begin{split}\int_{-\infty}^{+\infty}s_{u}\
p_{asym}(s_{u},0)ds_{u}&=2M_{11}=C_{12}\frac{1}{\sqrt{2\pi}}\left(\frac{m_{u}}{m_{c}}\right)^{1/2},\\\
\int_{-\infty}^{+\infty}\mathrm{sign}(s_{u})\
p_{asym}(s_{u},s_{c})ds_{u}ds_{c}&=4M_{11}=\frac{2}{3}C_{30}\frac{1}{\sqrt{2\pi}},\end{split}$
(23)

$C_{12}=2\sqrt{2\pi}\left(\frac{m_{c}}{m_{u}}\right)^{1/2}M_{11}\qquad,\qquad
C_{30}=6\sqrt{2\pi}M_{11}.$ (24)

Note that this implies that :

$C_{30}/C_{12}=3\sigma_{u}/\sigma_{c},$ (25)

a relation involving the directional slopes root mean squares
$\displaystyle\sigma_{u,c}=m^{1/2}_{u,c}$ which is to our best knowledge
original.

####  5.2.4 Estimation of the MSS shape

Useful quantities in optical and microwave remote sensing of the sea surface
are the so-called “MSS-shape” (Jackson et al., 1992; Chapron et al., 2000;
Nouguier et al., 2016), obtained by fitting the measured pdf with a Gaussian
distribution. They are primary parameters to describe the data, which are more
easily determined than $\displaystyle m_{u}$ and $\displaystyle m_{s}$ and
appear naturally when using the well-known Geometrical Optics model (Hagfors,
1966) in near-nadir scattering. Similarly to the up- and cross-wind MSSs, we
define an upwind ($\displaystyle m_{us}$) and a downwind ($\displaystyle
m_{cs}$) MSS-shape by:

$p(s_{u},s_{c})/p(0,0)\simeq e^{-\frac{1}{2}s_{u}^{2}/m_{us}}\
e^{-\frac{1}{2}s_{c}^{2}/m_{cs}}.$ (26)

Expansions of Eqs. (9) and (26) to second-order in slope and identification of
the quadratic terms shows that $\displaystyle m_{us}$ and $\displaystyle
m_{cs}$ are related to $\displaystyle m_{u}$ and $\displaystyle m_{c}$ by:

$\begin{split}m_{us}&=\left(1+\frac{C_{40}+C_{22}}{2+C_{04}/4+C_{40}/4+C_{22}/2}\right)^{-1}\
m_{u},\\\
m_{cs}&=\left(1+\frac{C_{04}+C_{22}}{2+C_{04}/4+C_{40}/4+C_{22}/2}\right)^{-1}\
m_{c}.\\\ \end{split}$ (27)

$\displaystyle m_{us}$ and $\displaystyle m_{uc}$ are thus in general smaller
than $\displaystyle m_{u}$ and $\displaystyle m_{c}$, respectively, with
differences resulting for the non-Gaussian nature of the distribution due to
its peakedness (which is found most often with a positive kurtosis). The MSS-
shape can be easily and robustly determined since they do not require an
absolute calibration of the probabilities and are quite insensitive to the
uncertainties affecting the tail of the pdf. Indeed, Eq. (26) implies that the
functions $\displaystyle s_{u}^{2}p(s_{u},0)/p(0,0)$ and $\displaystyle
s_{c}^{2}p(0,s_{c})/p(0,0)$ reach their maxima, of $\displaystyle
2m_{us}e^{-1}$ and $\displaystyle 2m_{uc}e^{-1}$, at $\displaystyle
s_{u}=\sqrt{2m_{us}}$ and $\displaystyle s_{c}=\sqrt{2m_{uc}}$, respectively,
which are only at 1.4 standard deviation away from $\displaystyle
s_{u}=s_{c}=0$.

##  6 Results and discussion

###  6.1 Results

The parameters feeding Eq. (9), determined as explained in Sec. 5.2 from the
probabilities retrieved from IASI observations, are given in Table 1. They
include the directional MSSs ($\displaystyle m_{u}$ and $\displaystyle
m_{c}$), the kurtosis coefficients ($\displaystyle C_{40}$, $\displaystyle
C_{04}$, and $\displaystyle C_{22}$) and the skewness ($\displaystyle C_{12}$
and $\displaystyle C_{30}$) coefficients. We also give the directional MSS-
shape ($\displaystyle m_{us}$ and $\displaystyle m_{cs}$). Note that no
results are given for wind speeds $\displaystyle U_{10}<3$ m/s as the observed
distributions for slower winds are too far from a Gaussian shape and could not
be properly described by a GC model, due to both an increased scatter of the
measured probabilities and a reduced number of observations. As shown by Fig.
7, the values in Table 1 and Eq. (9) enable a very satisfactory description of
the measured probabilities. In particular, the progressive change, with
increasing wind speed, of the position of the maximum of the alongwind
probabilities is well taken into account. This shift can be quantified by the
wave-tilt angle $\displaystyle\theta_{pmax}$ at the position of the maximum,
for which we provide wind-speed dependent values (Table 1). Note that the
latter are perfectly consistent with the fit made by Chapron et al. (2002)
(their Figure 1) after the data of CM.

$\displaystyle U_{10}$ | $\displaystyle 100m_{u}$ | $\displaystyle 100m_{c}$ | $\displaystyle 100m_{us}$ | $\displaystyle 100m_{cs}$ | $\displaystyle C_{40}$ | $\displaystyle C_{04}$ | $\displaystyle C_{22}$ | $\displaystyle C_{12}$ | $\displaystyle C_{30}$ | $\displaystyle\theta_{pmax}$
---|---|---|---|---|---|---|---|---|---|---
3.0 | 1.10 | 0.97 | 1.02 | 0.89 | 0.16 | 0.28 | 0.16 | 0.01 | 0.02 | 0.1
3.5 | 1.16 | 1.03 | 1.06 | 0.97 | 0.18 | 0.18 | 0.08 | 0.01 | 0.02 | 0.1
4.0 | 1.24 | 1.11 | 1.13 | 1.05 | 0.21 | 0.11 | 0.05 | 0.01 | 0.02 | 0.1
4.5 | 1.34 | 1.20 | 1.21 | 1.14 | 0.24 | 0.08 | 0.03 | 0.01 | 0.02 | 0.1
5.0 | 1.46 | 1.28 | 1.30 | 1.22 | 0.26 | 0.08 | 0.02 | 0.00 | 0.01 | 0.0
5.5 | 1.59 | 1.37 | 1.41 | 1.31 | 0.27 | 0.08 | 0.02 | -0.00 | -0.01 | -0.1
6.0 | 1.74 | 1.45 | 1.52 | 1.39 | 0.29 | 0.05 | 0.02 | -0.01 | -0.03 | -0.1
6.5 | 1.91 | 1.53 | 1.67 | 1.47 | 0.32 | 0.03 | 0.01 | -0.02 | -0.05 | -0.2
7.0 | 2.09 | 1.62 | 1.83 | 1.55 | 0.32 | 0.01 | 0.01 | -0.02 | -0.07 | -0.4
7.5 | 2.28 | 1.70 | 1.99 | 1.64 | 0.31 | 0.01 | 0.00 | -0.03 | -0.10 | -0.5
8.0 | 2.49 | 1.79 | 2.18 | 1.74 | 0.32 | -0.01 | -0.01 | -0.03 | -0.12 | -0.7
8.5 | 2.68 | 1.88 | 2.35 | 1.84 | 0.30 | -0.03 | -0.02 | -0.04 | -0.14 | -0.9
9.0 | 2.86 | 1.98 | 2.53 | 1.93 | 0.27 | -0.03 | -0.03 | -0.05 | -0.17 | -1.0
9.5 | 3.05 | 2.08 | 2.72 | 2.04 | 0.24 | -0.04 | -0.04 | -0.06 | -0.20 | -1.3
10.0 | 3.23 | 2.20 | 2.92 | 2.16 | 0.21 | -0.02 | -0.03 | -0.06 | -0.24 | -1.5
10.5 | 3.42 | 2.33 | 3.14 | 2.27 | 0.14 | -0.01 | -0.02 | -0.07 | -0.27 | -1.7
11.0 | 3.60 | 2.46 | 3.35 | 2.42 | 0.09 | -0.00 | -0.00 | -0.09 | -0.32 | -2.1
11.5 | 3.79 | 2.59 | 3.59 | 2.56 | 0.05 | -0.01 | 0.02 | -0.10 | -0.36 | -2.3
12.0 | 3.98 | 2.72 | 3.84 | 2.71 | 0.00 | 0.02 | 0.04 | -0.11 | -0.39 | -2.5
12.5 | 4.13 | 2.83 | 4.04 | 2.83 | -0.03 | -0.01 | 0.05 | -0.12 | -0.43 | -2.8
13.0 | 4.40 | 3.00 | 4.27 | 3.03 | -0.01 | 0.02 | 0.11 | -0.13 | -0.46 | -2.9
13.5 | 4.46 | 3.06 | 4.40 | 3.12 | -0.04 | -0.00 | 0.08 | -0.13 | -0.45 | -3.0
14.0 | 4.52 | 3.10 | 4.30 | 3.07 | -0.01 | -0.09 | 0.04 | -0.12 | -0.43 | -3.0
14.5 | 4.75 | 3.26 | 4.67 | 3.32 | -0.05 | -0.04 | 0.10 | -0.12 | -0.44 | -3.0
15.0 | 4.86 | 3.36 | 4.81 | 3.45 | -0.01 | -0.03 | 0.10 | -0.11 | -0.41 | -2.9
Table 1: Values of the up- ($\displaystyle m_{u}$) and cross-wind
($\displaystyle m_{c}$) MSSs and the GC coefficients versus the wind speed
$\displaystyle U_{10}$(m/s) at 10 m above the surface. Also given are the MSS-
shape ($\displaystyle m_{us},m_{cs}$), and the wave-tilt angle
($\displaystyle\theta_{pmax}$, in degree) at the maximum probability. The MSSs
are given in percent (i.e. multiplied by $\displaystyle 100$).

Figure 7: IASI-measured (red dots) along- and cross-wind slope distributions
of slopes for different wind speeds and the values (black lines) computed
using Eq. (9) with the values in Table 1.

A more detail look at the asymmetry can be taken by considering the values of
$\displaystyle p_{asym}(s_{u},s_{c})$ [see Eq. (13)] in the principal section
$\displaystyle s_{c}=0$ displayed in Fig. 8, which call for two remarks. The
first is that our model for the pdf again leads to a very satisfactory
agreement with the IASI-derived values for this relatively small effect of the
wind on the wave slopes. The second is that there is a qualitative evolution
when wind speed increases, with a change of sign and a switch from a positive
oscillation (below 5 m/s) to a negative one (above 6 m/s). An important
consequence of this is that the upwind pdf is larger (resp. smaller) than the
downwind one at smaller (resp. larger) wind speeds. We have no explanation for
this change, pointed out here for the first time, which indicates a
qualitative change in the asymmetry of wave facets as they evolve under
complex hydrodynamical processes and wind-surface interactions.

Finally, Fig. 9 shows the evolution of the directional MSS-shape with wind
speed and their comparison with the MSSs. Both are very close except in the
upwind direction at moderate wind speed (5-10 m/s), where the MSS-shape is
significantly smaller (by about $\displaystyle 10\%$) than the upwind MSS .
This is consistent with the evolution of the kurtosis coefficients (see Table
1) which reach their largest values in the upwind direction and this range of
wind speed. Recall that the peakedness of the distribution, which is
quantified by the (excess) kurtosis coefficients $\displaystyle C_{40}$,
$\displaystyle C_{04}$ and $\displaystyle C_{22}$, has been interpreted by
Chapron et al. (2000) as the effect of compounding of the random processes
influencing the sea-surface slopes through the variability of small-scale
slope variance from one patch to another due to short-term natural processes
such as wind gust or modulation by large wave groups. It thus seems that the
maximal variability of the MSSs occurs at moderate wind speeds, which are also
the most frequent. To check the consistency of the estimations we also display
in Fig. 9 the “reconstructed” MSS inferred from the MSS-shape through Eq.
(27). As can be seen, they are in excellent agreement with those estimated
from the moments, except at the largest wind speeds where small differences
appear, which indicates a reduced accuracy of either estimations.

Figure 8: Asymmetric component $\displaystyle p_{asym}$, in the principal
section $\displaystyle s_{c}=0$, determined, using Eq. (13), from the IASI
data (red dots) and values (black lines) computed using Eq. (9) with the
values in Table 1. Figure 9: Comparison of the directional MSSs (solid lines)
obtained with the IASI data using Eq. (18) and the MSS-shape’s (squares)
obtained with Eq. (26). Also displayed (circles) are the MSSs inferred from
the MSS-shape using Eq. (27), showing consistent results. Red and blue colors
refer to the up- and cross-wind values, respectively.

###  6.2 Comparisons with previous results

#### The mean square slopes

Figure 10 displays our $\displaystyle m_{u}$ and $\displaystyle m_{c}$ values
together with some parameterizations versus wind speed available in the
literature. Note that the wind speeds used by CM were measured at 12.5 m above
the sea surface, while the other studies use an altitude of 10 m, and were
thus converted at 10 m using a classical von Karman logarithmic profile, see
e.g. (Tennekes, 1973), which resulted in their multiplication by
$\displaystyle 0.98$ for all wind speeds. As can be seen, our results confirm
the anomalous behavior of those of Mermelstein et al. (1994) and Ebuchi and
Kizu (2002) noticed by Zhang and Wang (2010) and indicated by the comparisons
with the other values (Cox and Munk, 1954a; Bréon and Henriot, 2006; Lenain et
al., 2019) in Figs. 10. Furthermore, our values are fully compatible with the
linear law of CM in Eq. (12), since they fall within the uncertainties on the
latter. However, it must be reminded that the error bars given by CM (Cox and
Munk, 1954a) were obtained by calculating the overall RMS difference between
their MSSs estimates at each available wind speed (in their case, 25 different
values) and their associated linear fits, thus measuring the ability of the
latter to describe the data rather than the uncertainties of their individual
MSSs values. In contrast, our uncertainties are much smaller, thanks to the
huge number of observations used, and enable to point out a small but
statistically significant bias of the CM linear relationship for
$\displaystyle m_{u}$ at moderate wind speeds (5-7 m/s), where our values are
5-8% smaller. The possible origin of this small difference is further
discussed in Sec. 6.3.

Figure 10: Evolution with wind speed of the up- ($\displaystyle m_{u}$, left
panel) and cross-wind ($\displaystyle m_{c}$, right panel) MSSs. The black
circles represent our values (given in Table 1), while the red lines
correspond to the CM linear laws with associated error bars (Cox and Munk,
1954a). Also given are various other proposed parameterizations from the
literature, by Mermelstein et al. (1994) (green), Ebuchi and Kizu (2002)
(cyan), Bréon and Henriot (2006) (blue) and Lenain et al. (2019) (magenta).

#### The kurtosis coefficients

Figure 11 shows the variations of the kurtosis coefficients with wind speed.
The alongwind peakedness ($\displaystyle C_{40}$) ranges from $\displaystyle
0$ to $\displaystyle 0.32$ at moderate wind speed with maximal values reached
between 6 and 9 m/s, slightly negative values being obtained beyond 12 m/s. It
is thus smaller than the wind-independent (mean) values proposed by BH
($\displaystyle 0.4\pm 0.1$) and CM ($\displaystyle 0.23\pm 0.41$) but
compatible with the latter when uncertainties are taken into account. Our
$\displaystyle C_{22}$ and $\displaystyle C_{04}$ coefficients are very small
(between $\displaystyle 0$ and $\displaystyle 0.1$) with slightly negative
values for $\displaystyle C_{04}$ above 11 m/s. Our results here are thus
significantly different from those of both CM ($\displaystyle C_{22}=0.12\pm
0.06,C_{04}=0.4\pm 0.23$) and BH ($\displaystyle C_{22}=0.12\pm
0.03,C_{04}=0.3\pm 0.05$). However note that CM could only obtain a very
coarse estimation of the kurtosis coefficients based on fitting the parameters
entering in the azimuthal expansion of Eq. (10) with very large uncertainties
on the obtained results. Concerning BH, they found values of the same order as
those of CM but, in contrast, their upwind kurtosis coefficient,
$\displaystyle C_{40}$, is larger than their crosswind one, $\displaystyle
C_{04}$, which is consistent with our results. Their uncertainties are small,
but they were obtained by measuring the deviation of the fit to the data by
wind-independent values and not from a proper error analysis. In addition, a
closer inspection of the values in Fig. 5 of Bréon and Henriot (2006) shows
influences of wind speed in qualitative agreement with our results:
$\displaystyle C_{40}$ shows a maximum at moderate wind speed with
significantly lower values for both small and high wind speeds ;
$\displaystyle C_{04}$ and $\displaystyle C_{22}$ remain almost constant at
small and moderate wind speeds and their observed variations at higher wind-
speed are less trustworthy due to the reduced number of observations. We were
not able to evaluate the uncertainties on our determinations of the kurtosis
coefficients since there is no simple way to measure the dispersion of values.
However, the estimation of $\displaystyle C_{40}$, $\displaystyle C_{04}$, and
$\displaystyle C_{22}$ that we made using a GC fit (See Sec. 5.2.2), is highly
dependent on the assumed MSSs and small variations of the latter can induce
large changes of the former. To evaluate the impact of the assumed input MSSs,
we carried a second determination of the kurtosis coefficients by re-fitting
the IASI data while imposing the MSSs values of CM instead of our owns. As
shown by Fig. 11, this leads to a much larger value of $\displaystyle
C_{40}=0.6$ at 6 m/s, consistent with the findings of BH for this coefficient.
However, no significant changes were observed for $\displaystyle C_{04}$ and
$\displaystyle C_{22}$ which keep oscillating around zero. Note that slightly
negative kurtosis coefficients at high wind speed were also found in the
recent airborne experiments by Lenain et al. (2019).

Figure 11: Evolution of the kurtosis GC coefficients with wind speed, as
obtained from fits of the IASI data using the MSSs given by Eq. (22) (black
circles) or those of CM (red circles). The red dashed horizontal lines and the
blue dots are the wind-independent values proposed by CM and BH.

#### The skewness coefficients

Figure 12 shows the variations of the skewness coefficients, in which we
display the values obtained using the two available techniques (fit or moment)
to check the consistency of our estimations. As can be seen, the two
techniques lead to results that are in very good agreement, except beyond 13
m/s. This can be explained by the fact that the moment $\displaystyle{\cal
M\mit}_{11}$ cannot be estimated accurately for strong winds due to a slower
decrease of $\displaystyle a_{1}$ with the wave tilt. This is demonstrated by
computing the mean slopes, which should be zeros, i.e.:

$\begin{split}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}s_{u}p(s_{u},s_{c})ds_{u}ds_{c}=\pi{\cal
M\mit}_{21}=0.\end{split}$ (28)

The numerical evaluation of $\displaystyle{\cal M\mit}_{21}$ indeed shows that
it is of the order of $\displaystyle 10^{-4}$ for slow winds and of
$\displaystyle 10^{-2}$ for the largest wind speed (15 m/s), a growing
deviation from zero due to the increasing inaccuracy in the tail of the
measured distribution. Now regarding the comparison of our skewness
coefficients with those of CM and BH, Fig. 12 shows a good qualitative
agreement, all results being negative and steadily decreasing with increasing
wind speed. Furthermore, our results are fully compatible with those of CM
when the uncertainties on their linear laws are taken into account. This also
roughly stands for the difference between our results for $\displaystyle
C_{12}$ and those of BH, but not for $\displaystyle C_{30}$, the BH given
uncertainties on both coefficients being $\displaystyle\pm 0.01$. However, our
results support the BH observation of a non-linear relationship with wind
speed as opposed to CM. As for the kurtosis, we were not able to devise
meaningful uncertainties for the skewness coefficients. However, as done
before, we checked the influence of the MSSs by retrieving the skewness
coefficients from fits of the IASI data using the CM MSSs as inputs. The
results shown in Fig. 12 demonstrate a very good consistency between the
values retrieved using the two sets of MSSs.

Figure 12: Evolution of the skewness GC coefficients with wind speed as fitted
from the IASI data on the basis of the MSSs given by Eq. (22) (black circles)
or those of CM (red circles). Also shown are the coefficients obtained from
the first moment formula of Eq. (23) (black crosses) and the values proposed
by CM and BH (red and blue dashed lines).

###  6.3 Discussion

We here discuss, in view of the differences between our results and those of
CM pointed out in the preceding section, the two main hypotheses made by Cox
and Munk (1954a).

The first is that they assumed that the ’complete’ MSSs can be obtained by
applying a “blanket increase” to the ’incomplete’ ones (i.e, those obtained by
integration of the second moments up to the limiting slopes) regardless of the
wind speed [see Eq. (11)]. This rule was established on the basis of the
empirical fit using Eq. (10), which was assumed to hold beyond the limit of
the observed slopes, that is $\displaystyle
s_{4}=(4/\alpha_{0}^{\prime})^{1/2}$, where $\displaystyle\alpha_{0}^{\prime}$
is the quadratic slope coefficient appearing in Eq. (10). In Wentz (1976), the
variance renormalization procedure of CM was criticized on the ground that,
based on the logarithmic coefficients [Eq. (10), the upper bound
$\displaystyle s_{8}=(8/\alpha_{0}^{\prime})^{1/2}$ was not sufficient to
reach convergence of the integral [Eq. (8)] and was therefore arbitrary. We
have checked this assumption by numerical simulations based on the GC model
(not the logarithmic fit), and found that the integral Eq. (8) is well
converged at $\displaystyle s_{8}$ and therefore the renormalization factor
well-defined. However, contrarily to CM, we found the resulting factors
$\displaystyle R_{u}/R_{c}$ to be wind-speed dependent, where we have defined:

$R_{u}=\frac{\int_{|s|\leq
s_{8}}s_{u}^{2}p(s_{u},s_{c})ds_{u}ds_{c}}{\int_{|s|\leq
s_{4}}s_{u}^{2}p(s_{u},s_{c})ds_{u}ds_{c}},R_{c}=\frac{\int_{|s|\leq
s_{8}}s_{c}^{2}p(s_{u},s_{c})ds_{u}ds_{c}}{\int_{|s|\leq
s_{4}}s_{c}^{2}p(s_{u},s_{c})ds_{u}ds_{c}}.$ (29)

As shown by CM (Cox and Munk, 1956) by expanding the GC formula in power of
the wave tilt, one has:

$\alpha_{0}^{\prime}=\frac{1}{4}\left(\frac{1}{m_{u}}+\frac{1}{m_{c}}\right)+\frac{1}{8}\left(\frac{C_{40}}{m_{u}}+C_{22}(\frac{1}{m_{u}}+\frac{1}{m_{c}})+\frac{C_{04}}{m_{c}}\right).$
(30)

We then carried calculations of $\displaystyle R_{u}$ and $\displaystyle
R_{c}$ using three different input datasets: (D1) The MSSs and kurtosis and
skewness coefficients obtained (Table 1) from the analysis of IASI data. (D2)
The BH (Bréon and Henriot, 2006) MSSs and the associated kurtosis and skewness
coefficients. (D3) The CM (Cox and Munk, 1954a) values of the MSSs and
kurtosis and skewness coefficients. The resulting ratios $\displaystyle R_{u}$
and $\displaystyle R_{c}$ are shown in Fig. 13, calling for several remarks.
First, it is clear that none of the input datasets leads to results consistent
with the constant values $\displaystyle 1.23/1.22$ mentioned by CM. Secondly,
the ratio $\displaystyle R_{u}/R_{c}$ is very sensitive to the kurtosis
coefficients, as demonstrated by the important differences between the results
obtained using BH (D2) and CM (D3) parametrizations despite that the
associated MSSs are almost identical. In contrast, it is weakly dependent on
the MSSs, as seen around 6 m/s where the CM and IASI MSSs differ most while
the upwind kurtosis coefficients are close. If one assumes the correctness of
the $\displaystyle 1.23/1.22$ factors of CM, this means that these values can
only be reached using accurate and wind-dependent values of the kurtosis
coefficients corresponding to the specific sea states conditions encountered
in the CM experiments. To elucidate this issue, we determined the kurtosis
coefficients ($\displaystyle C_{40},C_{04}$) for which the computed values of
$\displaystyle R_{u}$ and $\displaystyle R_{c}$ are 1.23 and 1.22. To reduce
the number of unknowns we fixed the values of the MSSs and of $\displaystyle
C_{22}$, which has the smallest uncertainty ($\displaystyle\pm 0.06$), to
those of C.M The same exercise was carried out using the IASI MSSs. The
results are shown in Fig. 14 together with the mean values $\displaystyle
C_{40}=0.23,\ C_{04}=0.4$ found by CM. It turns out that the kurtosis
coefficients clearly vary with wind speed but have quite constant values at
small (3-6 m/s) and large (9-15 m/s) wind speeds, where the mean values
inferred by CM are respectively quite correct and very inaccurate. The use of
IASI MSSs leads to results that are closer to the $\displaystyle C_{40/04}$
measured by CM, a fact that supports the small negative correction brought by
IASI with respect to the CM linear law [Eq. (12)] for the MSS in the upwind
direction shown in Fig. 9. Another important outcome of this numerical test is
the occurrence of a negative upwind kurtosis ($\displaystyle
C_{40}\simeq-0.15$) at large wind speed, which tends to confirm the recent
observations by Lenain et al. (2019). Another specificity of the CM results is
their very large crosswind kurtosis $\displaystyle C_{04}\simeq 0.55$ . This
is clearly not representative of the average sea states observed with IASI
that have a negligible crosswind kurtosis and a more pronounced upwind one. We
hypothetize that this difference is specific to the wave conditions
encountered by CM in Hawaï.

Figure 13: Evolutions of the ’complete to incomplete’ variance MSSs ratios
$\displaystyle R_{u}$ (in red) and $\displaystyle R_{c}$ (in blue) calculated
using Eq. (29) and the datasets (see text) (D1), (D2) and (D3) represented by
circles, dashed and full lines, respectively. The dashed black line denotes
the constant factor $\displaystyle 1.23$ assumed by CM, see Eq. (11) Figure
14: Upwind ($\displaystyle C_{40}$, in red) and cross-wind ($\displaystyle
C_{04}$, in blue) kurtosis coefficients that best fulfill (see text)
$\displaystyle R_{u}=1.23$ and $\displaystyle R_{c}=1.22$, assuming CM MSSs
(solid lines) or IASI MSSs (squares). The dashed line denote the constant
values of CM.

The second important hypothesis made by CM is that of an homogeneous water
surface with homogeneous sky illumination. For this, the presence of natural
slicks has been neglected and the effect of cloud shadows were removed by
manual selection of the image photographs. A special attention has also been
given to the elimination of whitecaps, which correspond to dark spots on the
photometric plates. These were removed from the measurements by careful visual
inspection but the authors acknowledged that this was troublesome at large
wind speeds. In their data analysis, BH also eliminated the contribution of
foam and aerosols by rejecting the observations for which the off-glint
reflectance was larger than some threshold. This was made possible by the
multi-directional capability of the POLDER instrument, which could observe a
same point from different viewing angles within a short fly time of the
satellite. However, this is a very coarse criterion as the spatial resolution
of the satellite data (a few tens of km2) does not allow to resolve the foam
patches on the sea surface. Hence eliminating the measurements with the
highest off-glint reflectance does not exclude a contribution of whitecaps to
the remaining measurements, albeit smaller. Furthermore, this technique can
create a statistical bias in the slope distribution as it selects some
specific sea states at given wind speed (those with less whitecaps).
Nevertheless, despite an imperfect elimination of whitecaps in both the CM and
BH studies, the retrieved MSSs are in excellent agreement, indicating a
negligible impact of the foam leftover.

In our processing of the IASI measurement, cloudy and aerosol-contaminated
observations have been removed from the data set, and natural surface slicks
have been ignored. However, the configuration of the IASI instrument does not
allow to measure glint and off-glint reflectance at the same ground location
so that the white cap elimination procedure of BH could not be reproduced.
Hence, the whitecap contribution, if any, is included in the IASI radiances
but disregarded in our analysis for the retrieval of the wave-slope
probability (and sea surface temperature). Now, evaluating this perturbation
is an impossible task in the present state of knowledge. Indeed, due to the
dispersion in the whitecap coverage models [see e.g. the review by Anguelova
and Webster (2006)], to the complexity of the geometrical and statistical
description of breaking waves and foam [e.g. (Sharkov, 2007)], to the opposite
impacts on the radiometric parameters according to whether the foam is an
active or passive stage (Marmorino and Smith, 2005; Potter et al., 2015), and
to the increased contrast of sea foam reflectivity at large incidence angle
(Branch et al., 2016; Niclos et al., 2007), we have no reliable model at hand
to correct precisely for the influence of whitecaps.

However, in the mid-infrared range ($\displaystyle 3.6-4\mu$m) used in the
present study, the emissivity of sea water and foam have close values (Branch
et al., 2016), a statement that also applies to the reflectance [respectively
$\displaystyle\sim 2.5\%$ and $\displaystyle\sim 3\%$ at 4
$\displaystyle\mu$m, see Figure 5 of Salisbury et al. (1993)]. Furthermore,
the sea surface temperatures retrieved from IASI (Capelle et al., 2022;
Capelle and Hartmann, 2022) show an excellent agreement with in situ
measurements [taking into account the cool-skin effect (Fairall et al., 1996)]
with an overall flat RMS difference when plotted against wind speed,
suggesting that foam patches inside the field of view globally have a very
small influence on the brightness temperature around 4 $\displaystyle\mu$m.
This weak radiometric contrast indicates that the presence of breaking waves
have an effect on the sea surface radiance that is much smaller than in the
visible and near-infrared region, where foam and seawater reflectivity can
differ by one order of magnitude (Frouin et al., 1996; Dierssen, 2019). Given
the small spatial coverage of whitecaps for the considered wind speeds (a few
percents at 15 m/s wind speed, see e.g. (Anguelova and Webster, 2006; Albert
et al., 2016)) this can only cause minor change in the measured surface
radiance and therefore only impact the retrieval of very infrequent slopes. As
mentioned in Section 4, we believe that this partly explains the plateau
observed in the retrieved distribution at large slopes (see Figure 5). The
value of this offset defines the maximal available slope that can be obtained
through the Geometrical Optics approach. Recall that this maximal slope was
found larger by a few degrees than CM maximal slope, a finding which is
consistent with the reduced reflectivity of whitecaps (hence a lower plateau).
This bias has been corrected by subtracting a constant value that allows to
recover the normalization condition of the pdf and eliminate the mean whitecap
contribution. Even though this is a coarse correction, it is sufficient for an
accurate determination of the MSSs as those are evaluated through the lowest
moments of the distribution (which converge rapidly and have little
sensitivity to the tail). We therefore believe that there is no significant
contamination of the MSSs by whitecaps in our data processing. This explains
the good agreement between our values and those of CM, except perhaps for the
largest wind speeds (> 13 m/s). In their study using an airborne lidar on the
ocean and a laser glint counting technique, (Lenain et al., 2019) also
reached, despite the fact that they did not filter out the whitecaps, a good
agreement with CM and BH for the retrieved MSSs. Their skewness and kurtosis
coefficients, however, were very different from those of CM and BH and the
authors hypothetized that this might be due to the presence of whitecaps and
breaking. Since these coefficients were evaluated with a direct calculation of
the third and fourth moments of the distribution, which are very sensitive to
the tail of the distribution, their hypothesis is consistent with our
analysis. In our case, however, the higher-order Gram-Charlier parameters were
obtained through a least-square fitting of the symmetric and asymmetric part
of the distribution (once the MSSs obtained), a technique that prevents from
computing higher moments of the distribution and gives more weight to the
central part of the pdf. With this procedure, the accuracy of the retrieved
skewness and kurtosis is mainly conditioned by the accuracy of the MSSs since
any error in the latter can be partially compensated by changing the kurtosis
in the best fit.

The third assumption implicitly made by CM to obtain the laws in Eq. (12),
which is a major one, is that the wave-slope probabilities are only driven by
the speed and direction of the wind. Together with most previous studies,
including those of Bréon and Henriot (2006) and Ebuchi and Kizu (2002), our
analysis also used this hypothesis. However recall an important difference,
useful for the discussion made below, between the latter three investigations
and that of CM: In the former, the MSSs (and the other parameters of the GC
expansion) were obtained, for each wind speed, from very large numbers of
observations, while only one photograph was taken for each wind condition by
CM. This implies that, if a parameter other than the wind also monitors the
wave-slope probabilities, its influence in the retrieved pdf for a given wind
speed has been largely reduced by the averaging made in studies based on large
observation datasets, which is not the case for the CM study. Now, recall that
the measured probabilities show, from one observation to another, an important
scatter around the fitted pdf (see Figs. 5 and 6). Note that comparable
scatters were also obtained in the other studies involving great numbers of
data [see the error bars in the Figs. 3 of (Bréon and Henriot, 2006) and
(Ebuchi and Kizu, 2002)]. In our case, this dispersion could be due to several
effects, including: (E1) the noise on the IASI radiances and random errors in
the forward model used to determine (see Sec. 3) the probabilities; (E2)
inaccuracies in the assumed wind speeds and directions, which were provided by
the ECMWF/ERA5 reanalysis (see the first paragraph of Sec. 4) ; (E3) the
influence of meteo-oceanographic variables other than the wind on the wave-
slope statistics. A careful analysis of our results and procedures shows that
the first explanation, (E1), can be ruled out, at least for the large values
of the retrieved probabilities, typically for $\displaystyle p(s_{u},s_{c},U)$
above 1, which are accurately determined. Concerning the second explanation,
(E2), an error analysis shows that, in many cases, the observed scatter of the
probabilities is by far beyond what could result from the uncertainties on the
wind speed and direction. This is particularly the case near the inflection
point where $\displaystyle\partial p(s_{u},s_{c},U)/\partial U=0$, since the
probability is here practically insensitive to the wind but close to
$\displaystyle p(0,0,U)/e$ so that it is significant and thus well determined.
It thus appears that the third explanation, (E3), stands in many cases, and
the influences of the significant wave height, Richardson number, fetch, wave
age, etc, may be invoked. From this point of view, recall that some previous
studies point out the roles played by the wave height (Nouguier et al., 2016;
Guérin et al., 2017; Stopa et al., 2016), the swell (Hwang, 2008), and the
degree of atmospheric stability (Hwang and Shemdin, 1988; Shaw and Churnside,
1997). However note that the findings of these last two references were not
confirmed by Ross and Dion (2007) and Lenain et al. (2019) and that the
relative impact of swell on the MSS is still subject to some controversy Hwang
(2009); Hauser et al. (2009). Clarifying the contribution of each of the above
mentioned factors requires a substantial analysis which goes beyond the scope
of this paper. However, in order to quantify the effect of the scatter
observed in the present work on the individual values of the MSSs, we used an
isotropic approximation of the probability at the origin $\displaystyle
s_{u}=s_{c}=0$, $\displaystyle p_{0}\simeq 1/\pi m_{t})$ (with $\displaystyle
m_{t}\simeq 2m_{u}\simeq 2m_{c}$), from which we obtain $\displaystyle\Delta
m_{u/c}\simeq\pi m_{u/c}^{2}\Delta p$, where $\displaystyle\Delta p$ is the
standard deviation of the measured probabilities around $\displaystyle s=0$
(for $\displaystyle s<\epsilon$, where $\displaystyle\epsilon$ is taken to be
a small fraction of the RMS slope). This leads to values of
$\displaystyle\Delta m_{u/c}/m_{u/c}$ that decrease with wind speed and vary
between typically 16 % and 10 %. It is important to here emphasize that
$\displaystyle\Delta m_{u/c}$ quantifies the potential variability of the MSSs
obtained from one individual observation to another and not the uncertainty on
the values given in Table 1. Indeed, the latter are considerably more
accurate, since they were derived from huge numbers of data points, as
indicated by Fig. 5. It is now interesting to compare our MSSs with the CM
original data for a clean sea listed in Table I of Cox and Munk (1956). This
is done in Fig. 15 where the vertical bars around our values indicate the
range $\displaystyle\Delta m_{u/c}$ of variability of $\displaystyle m_{u/c}$
associated with the above discussed influence of meteo-oceanographic variables
other than the wind. As can be seen, the CM values full within this range so
that their unsmooth evolution with wind speed and their local differences with
our results can be explained by the fact that the wind over the observed sea
spot may not be the only parameter driving the wave-slope pdf. This reconciles
the two datasets and confirms their compatibility.

Figure 15: Original MSSs data derived by CM (Cox and Munk, 1956) (squares),
the linear laws deduced from the latter (dashed lines) and comparison with
those (circles) derived in the present work (Table 1), the error bars
indicating the range of variability. The blue and red colors refer to
$\displaystyle m_{c}$ and $\displaystyle m_{u}$, respectively.

##  7 Conclusion

We have determined the statistics of the slopes of the wave facets from space-
borne observations of the sea surface. This was done using a proper treatment
of the information brought by the contribution of solar photons to the
radiances collected, around 3.8 $\displaystyle\mu$m, by the Infrared
Atmospheric Sounder Interferometer (IASI). The obtained dataset was analyzed
using an original and robust approach providing the wind-dependent 7
parameters of a Gram-Charlier description of the wave-slopes probability
distribution function. The results are fully consistent, when uncertainties
are considered, with those of the pioneer study of CM (Cox and Munk, 1954a),
which so far remains the reference. However, thanks to the unprecedented
number (about 150 million) of IASI-retrieved probabilities and to the
excellent absolute calibration of the instrument, we were able to accurately
quantify the influence of wind speed, not only on the mean square slopes but
also on the kurtosis and skewness parameters, which CM could not do. We
pointed out a slightly reduced upwind MSS with respect to the CM linear law
with wind speed, a difference for which we provided two possible explanations:
The inadequacy of the linear fit in wind speed or specific sea states in the
CM experiments that differ from the mean conditions encountered in the present
study. The kurtosis and skewness coefficients were shown to follow a clear
tendency with wind speed, in qualitative but not quantitative agreement with
the last determinations by BH. The dominant kurtosis coefficient was found
upwind ($\displaystyle C_{40}$) with maximal values ($\displaystyle\simeq
0.3$) reached at moderate wind speeds while the other 2 coefficients
($\displaystyle C_{04},C_{22}$) were found quite constant and close to zero.
Non-linear wind-speed dependences were obtained for the skewness coefficients
but their absolute values were found smaller than those of BH.

We also pointed out and unexpected and so far undetected change of the sign of
the up/down wind asymmetry with growing wind speed that we cannot explain but
which certainly reveals some refined effects in the mechanisms governing the
wave facets creation.

Finally, the IASI-retrieved wave-slope probabilities for given values of the
wind speed and along- and cross-wind slopes show a significant scatter. A
preliminary analysis indicates that the latter is often beyond the
uncertainties on our determinations and may thus confirm that wind is not the
only variable monitoring the wave-slope pdf. Investigating this key issue and
quantifying the influences of other oceanic and atmospheric variables (the
significant wave height, the atmospheric stability, fetch, etc) is beyond the
scope of the present paper but it will be the subject of a forthcoming
investigation.

## Declaration of Competing Interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Acknowledgment

This work has been supported in part by CNRS, CNES and Ecole polytechnique. VC
and JMH acknowledge the IPSL mesocenter ESPRI facility for computer
simulations, as well as EUMETSAT and the Aeris infrastructure (Aeris) for
access to the IASI Level 1c data.

## Appendix: Forward model equations

Note that more details can be found in Capelle and Hartmann (2022), and that
the equations given below are written for the geometry of Fig. 1.

##### The surface-emission contribution Isurf

Assuming that the sea-surface directional emissivity
$\displaystyle\epsilon(\sigma,\theta_{I})$ does not depend on the SST
($\displaystyle T_{surf})$, and since both
$\displaystyle\epsilon(\sigma,\theta_{I})$ and the black-body radiance
$\displaystyle I_{BB}(\sigma,T_{surf})$ are constant over the narrow
instrument function $\displaystyle F_{I}(\Delta\sigma)$ of IASI, the first
term in Eq. (1) is given by:

$I_{surf}(\sigma,\theta_{I},T_{surf})=\epsilon(\sigma,\theta_{I})\tau(\sigma,\theta_{I})I_{BB}(\sigma,T_{surf}),$
(31)

with

$\tau(\sigma,\theta_{I})=\int_{-\infty}^{+\infty}F_{I}(\sigma-\sigma^{\prime})exp\left[-\int_{0}^{z_{Inst}}\frac{\alpha(\sigma^{\prime},z)}{\cos(\theta_{I})}dz\right]d\sigma^{\prime}.$
(32)

where $\displaystyle z_{Inst}$ is the altitude of IASI, and
$\displaystyle\alpha(\sigma,z)$ is the atmospheric absorption coefficient at
wave number $\displaystyle\sigma$ and altitude z.

##### The upward atmospheric-emission contribution $\displaystyle I_{atm}^{+}$

Using the same notations, the second term in Eq. (1), representing the
emission of the atmospheric column between the surface and the instrument, is
given by:

$\begin{split}&I_{atm}^{+}(\sigma,\theta_{I})=\int_{-\infty}^{+\infty}F_{I}(\sigma-\sigma^{\prime})\int_{0}^{z_{Inst}}\frac{I_{Loc}(\sigma^{\prime},z,\theta_{I})}{\cos(\theta_{I})}dzd\sigma^{\prime},\end{split}$
(33)

with

$\begin{split}I_{Loc}(\sigma,z,\theta_{I})&=I_{BB}\left[\sigma,T(z)\right]\alpha(\sigma,z)\exp\left[-\int_{z}^{z_{Inst}}\frac{\alpha(\sigma,z^{\prime})}{\cos(\theta_{I})}dz^{\prime}\right].\end{split}$
(34)

##### The downward atmospheric-emission contribution $\displaystyle
I_{atm}^{-}$

The third term in Eq. (1) is the emission of the atmosphere toward the surface
that is then reflected and transmitted up to the instrument. Its rigorous
calculation is computationally costly since it requires an integration over
all possible downward-emission directions, each weighted by the proper bi-
directional reflectivity. However, as demonstrated in Capelle and Hartmann
(2022), $\displaystyle I_{atm}^{-}$, which makes a small contribution to the
total radiance, can be precisely evaluated by considering a single direction
with a zenith angle $\displaystyle\theta_{down}$= 53∘. One thus has:

$\begin{split}I_{atm}^{-}(\sigma,\theta_{I})&=\int_{-\infty}^{+\infty}F_{I}(\sigma-\sigma^{\prime})\int_{0}^{+\infty}I_{Loc^{\prime}}(\sigma^{\prime},z,\theta_{down})\\\
&dz/\cos(\theta_{down})d\sigma^{\prime},\end{split}$ (35)

with

$\begin{split}&I_{Loc{{}^{\prime}}}(\sigma,z,\theta_{down})=\left[1-\epsilon(\sigma,\theta_{down})\right]exp\left[-\int_{0}^{z_{Inst}}\frac{\alpha(\sigma,z{{}^{\prime}})dz^{\prime}}{\cos(\theta_{I})}\right]\\\
&\times
I_{BB}\left[\sigma,T(z)\right]\alpha(\sigma,z)exp\left[-\int_{0}^{z}\frac{\alpha(\sigma,z^{\prime})}{\cos(\theta_{down})}dz^{\prime}\right].\end{split}$
(36)

##### The solar contribution Isun

Finally, the contribution of the reflected solar photons, denoted as
$\displaystyle I_{sun}$ in Eq. (1), is given by:

$\begin{split}&I_{sun}(\sigma,\theta_{I},\varphi_{I},\theta_{S},\varphi_{S})=\frac{\rho(\sigma,\theta_{I},\varphi_{I},\theta_{S},\varphi_{S})d\Omega_{S}}{4\cos^{4}(\theta_{W})}\\\
&\times\int_{-\infty}^{+\infty}F_{I}(\sigma-\sigma^{\prime})I^{0}_{Sun}(\sigma^{\prime})\exp\left[-\int_{0}^{z_{Inst}}\frac{\alpha(\sigma^{\prime},z)}{\cos(\theta_{I})}dz\right]\\\
&\times\exp\left[-\int_{0}^{+\infty}\frac{\alpha(\sigma^{\prime},z)}{\cos(\theta_{S})}dz\right]d\sigma^{\prime},\end{split}$
(37)

where $\displaystyle I_{sun}^{0}$$\displaystyle(\sigma)$,
d$\displaystyle\Omega_{S}$, $\displaystyle\theta_{w}$, and
$\displaystyle\rho$(…) respectively denote the (disk-averaged)
extraterrestrial radiance emitted by the sun, the solid angle within which the
sun is seen from Earth, the wave-tilt angle, and the bi-directional spectral
reflectivity of the surface. These last two quantities are given by:

$tg(\theta_{w})^{2}=\frac{\sin(\theta_{S})^{2}+\sin(\theta_{I})^{2}+2\sin(\theta_{S})\sin(\theta_{I})\cos(\varphi_{S}-\varphi_{I})}{\left[\cos(\theta_{S})+\cos(\theta_{I})\right]^{2}},$
(38)

and

$\rho(\sigma,\theta_{I},\varphi_{I},\theta_{S},\varphi_{S})=\rho(\sigma,\theta_{Inc})=\frac{R_{\parallel}^{2}(\theta_{Inc})+R_{\perp}^{2}(\theta_{Inc})}{2},$
(39)

with (assuming a unity value of the index of refraction of air)

$R_{\parallel}(\theta_{Inc})=-\frac{n_{SW}\cos(\theta_{Inc})-\cos(\theta_{Trans})}{n_{SW}\cos(\theta_{Inc})+\cos(\theta_{Trans})},$
(40)

and

$R_{\perp}(\theta_{Inc})=+\frac{\cos(\theta_{Inc})-n_{SW}\cos(\theta_{Trans})}{\cos(\theta_{Inc})+n_{SW}\cos(\theta_{Trans})},$
(41)

where $\displaystyle n_{SW}(\sigma)$ denotes the complex refractive index of
sea water and the incidence and transmission angles are given by:

$\cos(\theta_{Inc})=\sqrt{\frac{1+\sin(\theta_{S})\sin(\theta_{I})\cos(\varphi_{S}-\varphi_{I})+\cos(\theta_{S})\cos(\theta_{I})}{2}}.$
(42)

$\sin(\theta_{Trans})=\sin(\theta_{Inc})/n_{SW}(\sigma).$ (43)

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