In remote sensing applications, clouds are generally characterized by two properties: cloud optical thickness (COT) and effective radius of
water–ice particles (

The role and evolution of clouds in the ongoing climate change are still unclear. Their radiative feedback due to temperature rise or due to the indirect effect of aerosols is insufficiently understood, and they are known to contribute to the uncertainties in the future Earth climate (IPCC report, 2021). An accurate estimation of cloud properties is therefore very important for constraining climate and meteorological models, improving the accuracy of climate forecasting, and monitoring the cloud cover evolution. The instruments on board Earth observation satellites allow continuous monitoring of the clouds and aerosols as well as retrieval of their properties from a regional to a global scale.

The cloud properties are retrieved using the information carried by measurements of the reflected, emitted, or transmitted radiation by the
clouds. Two main optical cloud properties are generally retrieved: the cloud optical thickness (COT) and the effective radius of the water–ice
particles forming the cloud (

The abovementioned methods are subject to several sources of error. A moderate perturbation in the retrieved COT and

Considering the spatial variability of the cloud macrophysical and microphysical properties, the errors induced by the use of a homogeneous horizontal
and vertical cloud model have been found to depend on the spatial resolution of the observed pixel, the wavelength, and the observation and
illumination geometries (Kato and Marshak, 2009; Zhang and Platnick, 2011; Zinner and Mayer, 2006; Davis et al., 1997; Oreopoulos and Davies, 1998;
Várnai and Marshak, 2009). From medium- to large-scale observations greater than 1

At smaller scales, as considered here, errors due to IPA become more dominant. At this scale, pixels can no longer be considered infinite and
independent from their adjacent pixels. Radiative energy passes from one column to the others depending on the COT gradient. This leads to a decrease in
the radiance of pixels with large optical thickness and an increase in the radiance of pixels with small optical thickness, which tends to smooth the
radiative field and thus the field of retrieved COT (Marshak et al., 1995). As a result, it can lead to a large underestimation of the retrieved
optical thickness (Cornet and Davies, 2008). Adding to these effects, for off-nadir observations, the tilted line of sight crosses different
atmospheric columns with variable extinctions and optical properties, which tend to additionally smooth the radiative field (Várnai and Davies,
1999; Kato and Marshak, 2009; Benner and Evans, 2001; Várnai and Marshak, 2003; Fauchez et al., 2018). In the case of fractional cloud fields not
examined under nadir observations, the edges of the clouds cause an increase in the radiances for high viewing angles, which in turn increases the
value of the retrieved COT (Várnai and Marshak, 2007) while overestimating the retrieved

The assumption of a vertically homogeneous profile inside the cloud is also questionable. The vertical distribution of the cloud droplets is important
to provide an accurate description of the radiative transfer in the cloud (Chang, 2002) and obtain a more accurate description of the cloud
microphysics such as the water content or the droplet number concentration. For simplicity, classical algorithms assume a vertically
homogeneous cloud model. However, several studies have shown a dependence between the retrieved effective radius and the shortwave infrared (SWIR) band used. These
differences are explained by the non-homogenous cloud vertical profiles and by the different sensitivities of spectral channels due to
wavelength-dependent cloud particle absorption (Platnick, 2000; Zhang et al., 2012). Indeed, the absorption by water droplets being stronger at
3.7

In operational algorithms, the retrieval of COT and

In this paper, we present a method based on the optimal estimation method (Rodgers, 2000) to
also separately derive each type of uncertainty and apply it to the measurements of the airborne radiometer named the Observing System Including
PolaRization in the Solar Infrared Spectrum (OSIRIS), which was developed in the Laboratoire d'Optique Atmosphérique (Auriol et al., 2008). OSIRIS
is the airborne simulator of the 3MI (Multi-viewing Multi-channel Multi-polarization Imager), planned to be launched on MetOp-SG in
2024. It can measure the degree of linear polarization from 440 to 2200

We couple the multi-angular multi-spectral measurements of OSIRIS with a statistical inversion method to obtain a flexible retrieval process of COT
and

The aim of this paper is not to give an exhaustive overview of the possible errors concerning optical thickness and effective radius retrievals but to simply introduce a method to derive the different sources of uncertainties from a specific case of data acquired during an airborne campaign. Uncertainties due to error measurements and non-retrieved parameters, but also to the assumed forward model, are considered. If generalized to several cloudy scenes, the partitioning of the errors can help us to understand if and which non-retrieved parameters or forward models need to be optimized to reduce the global uncertainties of the retrieved cloud parameters.

This article is organized as follows. Section 2 describes the basic characteristics of OSIRIS and some essential details of the campaign CALIOSIRIS-2. In Sect. 3, a detailed description of the retrieval methodology is presented, including the mathematical framework needed to compute the uncertainties of the retrieved cloud properties. In Sect. 4, a case study of a liquid cloud is presented and analyzed. We assessed the magnitude of different types of errors, such as the errors due to measurement noise, the errors linked to the fixed parameters in the simulations, and the errors related to the unrealistic homogeneous cloud assumption. The multi-angular retrievals and uncertainties are compared with the results obtained by the classical mono-angular bispectral retrieval algorithms in Sect. 5. Finally, Sect. 6 gives a summary and some concluding remarks.

We use the new imaging radiometer OSIRIS. We will go through the main characteristics of the instrument and the airborne campaign CALIOSIRIS. More details about OSIRIS can be found in Auriol et al. (2008).

Spectral wavelengths of the VIS–NIR

OSIRIS (Observing System Including PolaRization in the Solar Infrared Spectrum) is an extended version of the POLDER radiometer (Deschamps et al.,
1994) with multi-spectral and polarization capabilities extended to the near-infrared and shortwave infrared. This airborne instrument is a prototype of the
future spacecraft 3MI (Marbach et al., 2015) planned to be launched on MetOp-SG in 2024. It consists
of two optical sensors, each one with a two-dimensional array of detectors: one for the visible and near-infrared wavelengths (from 440 to
940

OSIRIS has eight spectral bands in the VIS–NIR and six in the SWIR. Similar to the concept of POLDER, OSIRIS contains a motorized wheel rotating the
filters in front of the detectors. The step-by-step motor allows only one filter to intercept the incoming radiation at a particular wavelength. The
polarization measurements are conducted using a second rotating wheel of polarizers. Given the sensor exposure and transfer times, the duration of a
full lap is about 7

OSIRIS is an imaging radiometer with a wide field of view. It has a sensor matrix that allows the acquisition of images with different viewing
angles. The same scene can thus be observed several times during successive acquisitions with variable geometries. The largest dimension of the sensor
matrix is oriented along-track of the aircraft to increase the number of viewing angles for the same target. For example, when the airplane is flying
at a 10

OSIRIS participated in the airborne campaign CALIOSIRIS in October 2014. It was carried out with the contributions of the French laboratories LOA (Laboratoire d'Optique Atmosphérique) and LATMOS (Laboratoire ATmosphères, Milieux, Observations Spatiales, Paris) and with Safire, the French Facility for Airborne Research. One objective of this campaign was the development of new cloud and aerosol property retrieval algorithms in anticipation of the future space mission of 3MI intending to improve our knowledge of clouds, aerosols, and cloud–aerosol interactions.

Studied case on 24 October 2014 at 09:02 UTC (11:02 LT, local time).

The data used in this work focus on a cloudy case over the ocean surface: a marine monolayer cloud that was observed on 24 October 2014 at 11:02 LT
(local time). The aircraft flew at an altitude of 11

The clouds backscatter total solar radiation more intensely in the cloudbow regions near 140

At the time of the CALIOSIRIS campaign in 2014, the polarized channels presented calibration and stray light issues, which make use of the polarized
measurements difficult for quantitative retrievals. In addition, the images from the two sensors were not well colocated. Consequently, for this
work, we use two unpolarized channels of the SWIR matrix, one almost non-absorbing (1240

In order to use the multi-angular capability of OSIRIS, successive images have to be colocated. After subtracting the average of similar successive images to remove the angular effects, the colocation is achieved by minimizing the root mean square difference of the radiances between each pair of successive images for different translations along the line and the column in the second image. The reference image is the central one of the sequence. Images with translations beyond the dimensions of the central image are ignored. Multi-angular radiances at the cloud level correspond in our case to 9 to 13 directions.

One of the most robust approaches in cloud property retrievals is the optimal estimation method (OEM). It is increasingly used in satellite measurement inversion (Cooper et al., 2003; Poulsen et al., 2012; Walther and Heindiger, 2012, Sourdeval et al., 2013; Wang et al., 2016). It provides a rigorous mathematical framework to estimate one or more parameters from different measurements. The OEM also characterizes the uncertainty in the retrieved parameters while taking into account the instrument error and the underlying physical model errors. A complete description of the optimal estimation method for atmospheric applications is given by Clive D. Rodgers (Rodgers, 2000). In this book, Rodgers exhaustively described the information content extraction from measurements, the optimization of the inverse problem, and the solutions and error derivations. In the following, we will go through the basics of this method that define the core of our retrieval algorithm.

Considering a vector

The OEM aims to find the best representation of parameters

where

In the optimal estimation method, the previous PDFs are represented by Gaussian distributions, assuming that the errors of the measurements, the errors related to the non-retrieved parameters, and the errors of the forward model are normally distributed around a mean value.

Therefore, it can be easily shown that the best estimate of the state vector

The first term of

The minimization is done through the Levenberg–Marquardt approach (Marquardt, 1963; Levenberg, 1944) based on the “Gauss–Newton” iterative
method. Assuming the model is nearly linear around a given state vector, each iteration is calculated following Eq. (

The parameter

The iterative process stops when the simulation fits the measurement (Eq.

When neither the inequality of Eq. (6) nor the inequality of Eq. (7) is reached after 15 iterations, the retrieval is considered a failed retrieval.

In order to apply this theoretical framework to our retrieval algorithm, we next define the basic elements stated in the previous subsection.

The state vector

It can be noted that because the relationship between radiances and optical thickness has a logarithmic shape, using

The a priori state vector was set to [10, 10

The measurement vector

The forward model is based on the adding–doubling method (De Haan et al., 1987; Van de Hulst, 1963) to solve the radiative transfer equation and simulate the radiances measured by OSIRIS for the corresponding observation geometries and wavelengths. It is a major element of the retrieval and describes the radiation interaction with the cloud, the surface, and the atmosphere while fixing several parameters (e.g., wind speed, cloud altitude). We assume a standard atmosphere with a midlatitude summer McClatchey profile (McClatchey et al., 1972) for the computation of molecular scattering. As the two channels used in the retrieval are in atmospheric windows (as seen in Fig. 1), the atmospheric absorption is not accounted for. It is not completely true, and therefore the cloud optical thickness will be slightly underestimated and the effective radius slightly overestimated. Our case study is purely above an ocean surface. The reflection by the surface can affect the measured radiances even in cloudy conditions and particularly for optically thin clouds. The anisotropic surface reflectance of the ocean surface is characterized by a bidirectional polarization distribution function (BPDF). We used the well-known Cox and Munk model to compute the specular reflection modulated by ocean waves (Cox and Munk, 1954) with a fixed ocean wind speed based on the NCEP reanalysis of the National Oceanic and Atmospheric Administration (NOAA).

As in current operational algorithms, the cloud model used for the retrieval is a plane-parallel and homogeneous (PPH) cloud, which implies the
independent pixel approximation (IPA). The case study is a liquid water cloud scene. Therefore, we used a lognormal distribution for the size of
particles, which are assumed to be spherical (Hansen and Travis, 1974) and described by an effective radius and an effective variance (

The Jacobian matrix

During the retrieval process, every element is associated with a random or systematic error embedded in the error covariance
matrix

In this formulation, we have assumed that the two terms of the state vector are independent, and thus the off-diagonal terms of

Besides the fixed parameters, the cloud model used in the radiative transfer computation can also be a source of uncertainty. The uncertainties of the
retrieved parameters related to this approximation are regrouped in the covariance matrix

Previous studies (Wang et al., 2016; Iwabuchi et al., 2016; Poulsen et al., 2012; Sourdeval et al., 2015) have already computed and presented the
uncertainties of the retrieved cloud properties for all error contributions using

The total variance–covariance matrix of the retrieved state vector (

Each term in this equation is developed and discussed in the following three subsections.

Any type of measurement is subject to errors. It is necessary to apply calibration processes to study the relationship between the electrical signals measured by the detectors and the radiances and quantify its uncertainty. Calibration is done during laboratory experiments before the airborne campaign or the instrument launch into space (Hickey and Karoli, 1974). It can be done in situ if calibration sources are available on board the sensor (Elsaesser and Kummerow, 2008) or it can be vicarious (e.g., Hagolle et al., 1999) by using natural or artificial sites on the surface of the Earth. The uncertainties of the measurements remaining after the calibration processes are assumed, random, and uncorrelated between channels and can be consistently approximated by a Gaussian probability density function over the measurement space.

As errors between measurements are supposed to be independent, the covariance matrix of measurement noise (

The error covariance matrix for the retrieved parameters due to measurement errors is then expressed by mapping the covariance
matrix

The uncertainty in a particular parameter

Any retrievals from remote sensing observations require prior knowledge of several unknown parameters used in the forward model computation. Those
parameters are not retrieved due to a lack of sufficient information. To compute the fixed parameters (fp) errors, we quantified the possible error in
our estimation of the fixed model parameters. In our case study, these parameters are the altitude of the cloud (alt), the effective variance of the
cloud droplet size distribution (

Each column in

Each covariance matrix from the right side of Eq. (

In order to develop

The last elements needed to resolve Eq. (

The values and the uncertainties of these fixed parameters are chosen according to the experimental setup of the campaign. To estimate the
uncertainties originating from the fixed cloud altitude, we used the opportunity of having the lidar-LNG aboard the aircraft, which gives the
backscattering signal obtained around the case study of CALIOSIRIS. From 11:01:06 to 11:03:06 CEST (the time when the same
cloud scene is apparent), the cloud altitude varies between 5.57 and 5.73

Concerning the effective variance

For the ocean wind speed fixed to 8

Forward models are usually formulated around some limitations and assumptions that can contribute to the uncertainty of the retrieved parameters. The
forward model used to simulate the radiances measured by OSIRIS follows the cloud plane-parallel assumption. This assumption is known to cause errors
in the retrieved parameters (see Sect. 1) that can be assessed and included in the total uncertainty. The evaluation of these modeling errors requires
an alternative forward model

The simplified model used for the retrieval can lead to biased retrieved parameters. In this case, the bias due to the model will be included in the Gaussian PDF width, resulting in an overestimation of the uncertainties.

The uncertainties related to the cloud vertical homogeneity and the cloud horizontal homogeneity are quantified separately. In the following, we present the elements of the forward model used to quantify the uncertainties of these assumptions.

The vertically heterogeneous cloud model to assess the uncertainties of the assumed homogeneous cloud model is described by

an effective radius profile and possibly an effective variance profile but for simplification – we will consider

an extinction coefficient (

a cloud geometrical thickness (CGT) characterized by the difference between the altitude of the cloud top (

The effective radius and extinction coefficient profiles are computed using an analytical model already introduced in Merlin (2016). It is based on adiabatic cloud profiles, which are described and used in several studies (Chang, 2002; Kokhanovsky and Rozanov, 2012). In the adiabatic scheme, the effective radius increases with altitude. However, several studies proved that a simple adiabatic profile is not sufficient to describe a realistic cloud profile (Platnick, 2000; Seethala and Horváth, 2010; Nakajima et al., 2010; Miller et al., 2016). Depending on the maturity of the cloud, turbulent and evaporation processes can reduce the size of droplets at the top of the cloud and/or collision and coalescence processes can increase the size of the droplets in the lower part of the clouds as observed by Doppler radar (Kollias et al., 2011). The profile used in this study aims to represent the case of droplet size reduction at the top of the cloud, but other and more sophisticated and representative profiles can be used (Saito et al., 2019).

The description of this more realistic vertical cloud profile is obtained with two adiabatic profiles (Fig. 3) that are joined at the altitude of
maximum liquid water content (LWC) called

The first profile from

The second profile from

The heterogeneous vertical profile of effective radius (black line) and extinction coefficient (blue line) used to assess uncertainties due to the assumption used for the vertical profile. The equivalent homogeneous vertical profiles are shown in dashed lines. The cloud is between 5 and 6

Considering that LWC is equal to zero at the base and top of the cloud and relying on the linear variation model of the LWC with altitude (

The profiles of effective radius (Eq. 23) and extinction coefficient (Eq. 24) can then be computed by considering that the particle concentration is
constant over the entire cloud, which makes it possible to obtain analytical functions of LWC,

A form factor

This unitless parameter

To assess the error due to the vertical heterogeneity of the cloud, we need to specify the maximum value of the extinction coefficient

A vertically heterogeneous cloud is computed for each pixel using the retrieved value based on the homogeneous assumption. The error covariance matrix
describing the error due to the simple homogenous cloud assumption (Eq. 21) is calculated from the difference between radiances computed with
homogeneous and heterogeneous vertical profiles, denoted

The other assumption that might affect the retrieved cloud optical properties in the current operational algorithms is the horizontally plane-parallel
and homogeneous (PPH) assumption for each observed pixel. It implies that each pixel is horizontally homogeneous and independent of the neighboring
pixels, known as the independent pixel approximation (IPA). The homogeneous PPH assumption affects the cloud-top radiances and leads to differences
between 1D and 3D radiances that are the result of several effects discussed in numerous publications and briefly summarized in Sect. 1. This PPH
assumption includes errors known as the PPH bias due to the subpixel variations of the cloud and errors related to the photon horizontal transport
between columns (IPA error). At the high spatial resolution of OSIRIS (less than 50

To assess the uncertainties in the retrievals arising from this assumption, Eq. (

Our strategy to assess the different types of uncertainty follows two steps. In a first step, we retrieve COT and

It should be noted that the parameters retrieved in the first step may be biased, in particular due to the use of a simplified cloud model to connect the state vector to the measurements. We assume that the estimation of the uncertainties performed in the second step is, however, correct if the variations predicted by the simplified and the realistic models around the retrieved values (potentially biased) and around the true values are identical. This is correct with a linear forward model but can be an assumption that is too strong in cloud retrieval regarding the nonlinearity of the relationship of the radiances as a function of cloud parameters. A way to test this assumption would be to use numerical experiments.

COT

In Fig. 4, COT (Fig. 4a) and

Uncertainties (RSD) in percent for COT

As detailed in Sect. 3.3, the final error is divided into three categories. Figure 5 shows the uncertainties originating from a 5 % measurement
error in the retrieved COT, RSD COT (mes), and in the retrieved

The uncertainty RSD (%) of COT (left column;

The second type of uncertainty is related to the fixed parameters in the forward model. In Fig. 6, we show the uncertainty in COT and

Figure 6c and d represent the uncertainties of the retrieved COT and

Figure 6e and f show that an error in the estimation of the ocean wind speed affects the retrieved
COT and

We note that all the uncertainties of the studied fixed parameters remain below 1 %, which shows that retrieval of all the COT–

The uncertainties (%) in COT and

The uncertainties due to the assumptions of the forward model are presented in Fig. 7. Figure 7a and b represent the uncertainties of COT and

The uncertainties originating from the use of a 1D radiative transfer code instead of a more realistic 3D radiative transfer are represented in
Fig. 7c and d for COT and

Considering the solar zenith incidence angle (59

The simulated 1D

At this scale, the effects related to the independent pixel approximation (IPA) (Oreopoulos and Davies, 1998) are dominant since the horizontal
transfers of photons between pixels are important. The smaller the column horizontal sizes that are considered, the more the real behavior of radiation in
the atmosphere will be misrepresented. The horizontal radiation transport (HRT) tends to smooth the radiative field by increasing or decreasing the
radiances according to the optical thickness gradient between the considered pixel and its neighbors. This effect is shown in Fig. 8. Figure 8b
and d, representing the reflectances computed with 3DMCPOL at 1240 and 2200

Histograms of the relative difference between the reflectances computed in 1D and 3D at 1240

In Fig. 9, histograms of the relative difference between the radiances computed in 1D (R1D) and the radiances computed in 3D (R3D) at
1240

Overall, we note that the uncertainties due to the forward model assumption are much more important than the ones due to the fixed parameters. The retrieval is not sensitive to small variations in the fixed parameters. However, while assessing uncertainties due to the vertical profile or radiative transfer assumption, we change the parameters that our forward model is proven to depend on, and thus changes in the integrated profile can lead to relatively large variations in the radiance fields and consequently large uncertainties.

The same strategy applied in Sect. 4 is applied using the bispectral mono-angular method used for the MODIS instrument. For the
mono-angular bispectral approach, the measurement vector

COT

The results are presented in Fig. 10. The retrieved COT over the whole field varies between 1 and 12 with a mean value equal to 3.44. Compared to
multi-angular measurements (mean COT of 2.13), the retrieved COT values tend to be higher. The range of retrieved

Uncertainties in the effective radius originating from the measurement errors, RSD

Multi-angular retrieval presents the major advantage that no aberrant values of

Except in the case of failed retrievals that occur for values outside the LUT ranges, the relation between radiances and COT–

Bar chart of the mean uncertainties of the retrieved COT and

To compare the uncertainties of the two retrievals, we use the relative standard deviation (RSD) to be consistent with the previous results. In
Fig. 12, we present the spatial average of the different types of errors, presented in Sect. 4, for the mono-angular method (light green for COT and
light blue for

Overall,

In Fig. 12b, for mono-angular retrieval, the measurement errors contribute to an uncertainty of
about 8 % in the retrieved COT and about 13 % in the retrieved

The following two groups of bars correspond to the errors introduced by the cloud homogeneous assumption used in the forward model. They are the main
source of errors. For mono-angular retrieval, the assumption of a vertical homogeneous profile contributes to an uncertainty of about 16 % in COT
and 54 % in

The multi-angular approach provides additional information for each pixel and constrains the forward model to match all the angular radiances at
once. As seen, the OSIRIS multi-angular characteristics have the advantage of decreasing the angular effects around the cloudbow directions by adding
the contribution of other geometries and mitigating the sensitivity of the retrieval issued from the assumptions in the forward model. It avoids most
of the failed convergences that occurred with the mono-angular bispectral method and retrieved more homogeneous and coherent COT and

In this study, we present a method to retrieve two important microphysical and optical parameters of liquid clouds, COT and

The studied case uses the measurements of the airborne radiometer OSIRIS obtained during the CALIOSIRIS campaign. It consists of a monolayer water
cloud located at 5

In the first step of the retrieval, COT and

In the multi-angular retrieval, a 5 % measurement error contributes to around 3 % uncertainty in the retrieved COT and 6 % in the
retrieved

The uncertainties related to the fixed parameters remain low with both mono- and multi-angular retrieval. The largest one is due to the unknown value
of the effective variance of the droplet size and is respectively equal to 0.15 % and 0.25 % for the mono- and multi-angular cases. Note that,
since the information provided by lidar or polarized measurements was used, the uncertainty for the non-retrieved parameters was chosen to be low. For
applications to cases without this available information, errors would be higher. If the method is applied to 3MI for example, the errors related to
the cloud-top altitude would be higher as the O

This study clearly shows that the largest uncertainty is due to the homogeneous cloud assumption made in our forward model. First, the uncertainties
related to the homogeneous vertical profile were quantified using a heterogeneous LWC profile with a triangle shape (known as quasi-adiabatic)
composed of two adiabatic profiles. This more realistic profile takes into account the transition zone at the top of the cloud related to turbulent
and evaporation processes. The scene-averaged values reach 5 % and 13 % for COT and

The other sources of uncertainty related to the simplified cloud physical model come from the radiative non-independence of the cloudy columns that
dominate at the high spatial resolution of OSIRIS. In the optically thin overcast cloud case studied here, the scene average uncertainties originating
from the 3D effects are 4 % for COT and 9 % for

The method was applied to real data, which means that the true cloud parameters are unknown. Consequently, it is not possible to know if real errors in the retrieved parameters are included in the uncertainties given by the method presented here. One factor that can lead to an erroneous assessment is that the estimations of the uncertainties are done around retrieved values that can be biased. A way to check the consistency of the method and the validity of the uncertainty ranges would be to simulate radiances using a large eddy simulation model with a realistic cloud physical description, add noise for the error measurements, and derive the cloud parameters and their uncertainties.

The method presented here can be adapted to the future 3MI. The first step, which consists of including the uncertainties related to the measurement errors, is directly implementable in an operational algorithm. The second step, which consists of computing the uncertainties resulting from the non-retrieved parameters, is more computationally expensive but could also be included. The uncertainties related to the non-retrieved parameters, in addition to the one related to measurement errors, have already been implemented since Collection 5 in the MODIS operational algorithm through the computation of a covariance matrix wherein Jacobians are derived from lookup table and completed for Collection 6 (Platnick et al., 2017). Concerning the forward model errors, the method cannot be implemented as in this work in an operational algorithm because of the prohibitive computation time, but a climatology based on several case studies, depending on the type of clouds, a land or ocean surface flag, for example, could be used in order to obtain a distribution of the errors according to the scene characteristics.

The results obtained in this study show, not surprisingly regarding the numerous studies already published, that the vertical and horizontal homogeneity assumptions are major contributors to the retrieval uncertainties. One way to reduce it would be to define a more complex cloud model that can take into account the vertical and horizontal heterogeneity. This adds more complexity to the forward model as it would imply retrieving more sophisticated cloud parameters (e.g., extinction or effective size profile). It appears, however, possible given the important and complementary information provided by OSIRIS or 3MI measurements. Recent studies proposed retrieving vertical profiles using cloud-side information (Ewald et al., 2018; Saito et al., 2019, Alexandrov et al., 2020) or realizing multi-pixel retrieval to account for the non-independence of the cloudy pixels (Martin et al., 2014; Okamura et al., 2017; Levis et al., 2015), and their implementation could be studied.

Data and codes are available upon request to the authors.

CM developed the inversion algorithms and performed the analysis of the results with the support of CC, FP, and LCL. FA and FP participated in the CALIOSIRIS airborne campaign. FA and JMN made the calibration and OSIRIS data processing from level 0 to level 1. All authors discussed the results and contributed to the final paper.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors thank Guillaume Merlin and Anthony Davis for fruitful discussions concerning the cloud vertical heterogeneous profile. Airborne data were obtained using the aircraft managed by Safire, the French Facility for Airborne Research, an infrastructure of the French National Center for Scientific Research (CNRS), Météo-France, and the French National Center for Space Studies (CNES).

This work has been supported by the Programme National de Télédétection Spatiale (PNTS,

This paper was edited by Alexander Kokhanovsky and reviewed by two anonymous referees.