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Introduction
Cirrus clouds cover from 15 % to 40 % of the Earth's surface (Sassen et al., 2008).The temperature difference between the cloud top and the surface leads to a warming of the atmosphere by capturing a part of the infrared radiation emitted by the Earth's surface.On the contrary, a part of the solar incident radiation is reflected to the space by the parasol effect, but this is generally slight for high clouds.Therefore, cirrus clouds lead to a positive radiative forcing (e.g. a greenhouse effect) and their knowledge and evolution are crucial in the understanding of the Earth's radiative budget (Hartmann and Short, 1980;Ohring and Clapp, 1980;Stephens, 2005;Eguchi et al., 2007).Introduction

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Full Global observations are well adapted to follow and better understand cloud evolution and characteristics.With this aim, many satellites are dedicated to their observations from visible to infrared wavelengths.Algorithms usually used to retrieve cloud parameters from passive instruments, such as optical thickness and effective diameter of ice crystals, assume that clouds are homogeneous and infinite between two planes.This assumption is called the homogeneous Independent Pixel Approximation (IPA, Cahalan et al., 1994) or Independent Column Approximation (ICA, Stephens et al., 1991).However, real clouds can be far from this idealized model and this assumption may lead to bias on the retrieval of clouds properties.
In this context, radiative transfer modelling is very useful to study the 1-D bias in function of the cirrus structure and composition.Many studies have been conducted on the impact of cloud heterogeneities in the visible range and principally for warm clouds (Marshak and Davis, 2005).However, only few studies have been performed on cirrus cloud heterogeneities in the thermal infrared and they concern mainly the fluxes or heating/cooling rates.Concerning the fluxes, Hogan and Kew (2005) showed that radiative transfer calculations using IPA can change the mean TOA radiative fluxes of about 45 W m −2 in the shortwave and 15 W m −2 in the longwave.Furthermore, Chen and Liou (2006) showed that significant impact exists on the broadband thermal cooling rates (around 10 %) when the 3-D radiative transfer is compared to 1-D radiative transfer.
As far as we know, no study has been made concerning the heterogeneity bias on the infrared radiative quantities measured by space sensors.However, satellites, such as the Imaging Infrared Radiometer (IIR, Garnier et al., 2012Garnier et al., , 2013) ) or the Moderate Resolution Imaging Spectroradiometer (MODIS, Cooper et al., 2007;Wang et al., 2011), use TOA Brightness Temperatures (BT) in the thermal infrared window to retrieve cloud parameters.In this paper, we study the impact of cirrus heterogeneities in this spectral domain.In Sect.2, we present the model 3DCloud (Szczap et al., 2013) used to generate realistic cloud scenes and the Monte-Carlo 3-D radiative transfer code named 3DMCPOL (Cornet et al., 2010)  three dimensional (3-D) atmospheres.We simulate BT for several cirrus generated from realistic conditions as well as from measurements made during the CiRus CLoud Experiment-II (CIRCLE II) airborne campaign.In Sect.3, the biases due to heterogeneities are quantified by comparing the 3-D and 1-D BT at the IIR spatial resolution (1 km × 1 km).Summary and conclusions are given in Sect. 4.

Cirrus cloud generation
In order to simulate the impact of cirrus heterogeneities on the TOA BT, realistic 3-D cirrus need to be generated.Firstly, 3-D cirrus Ice Water Content (IWC) was simulated with a cloud generator based on basic atmospheric equations as well as Fourier transform framework to constrain invariant scale properties.Then, the optical properties are parametrized with two different models described hereafter.

3-D Ice Water Content generation
Cirrus clouds are generated by the 3DCloud model (Szczap et al., 2013).Firstly, basic atmospheric equations with idealized meteorological profiles are resolved in order to simulate the 3-D IWC.Secondly, scale invariant properties are constrained by the iterative Fourier framework.Hogan and Kew (2005) have shown that the IWC or 3-D extinction are characterized by a power spectra with a −5/3 spectral slope.Generally, this spectral slope is delimited by a large scale limit L out and a smaller scale limit corresponding to the cloud pixel spatial resolution.Hogan and Kew (2005) estimated from radar reflectivity and cirrus temperature that the IWC spectral slope is equal to −5/3 from scales of the order of a meter to a L out of 50 km at the top of the cirrus, but it can decrease with the optical depth.Hogan and Kew (2005) have supposed that this decrease can be due to the coupled action of the wind shear with a spread of particle fall speeds leading to a homogenization of the IWC preferentially at smaller scales.With this behaviour, the cirrus must be old enough for that an important sedimentation Introduction

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Full of crystals to appear.However, as we show later on LIDAR measurements, we do not observe a change in the spectral slope with the altitude (see at the end of the section).Therefore, in our cirrus simulations, the spectral slope is assumed to be equal to −5/3 at all scales and altitudes.For this study, two different cloud structures are generated.The first cirrus field (Fig. 1) is based on meteorological profiles to form a cirrus cloud as presented by Starr and Cox (1985), with the addition of a wind profile to form virgas. From this first realisation, the influence on TOA BT of the cirrus mean optical thickness τ c , the cirrus heterogeneity parameter ρ τ , the ice crystal effective diameter D eff and the cirrus altitude are easily tested.The heterogeneity parameter is defined by Szczap et al. (2000) as ρ τ = σ τ /τ c , with σ τ the standard deviation of the optical thickness.Eight cirrus with different mean cloud parameters are generated (see Table 1).The cirrus mean optical thickness τ c increases from 0.45 to 1.8, the cirrus heterogeneity parameter ρ τ from 0.7 to 1.5, the ice crystal effective diameter D eff from 9.95 µm to 40.58 µm and the altitude from 7.97 km to 11.06 km.These macrophysical parameters cover the characteristics of usual cirrus clouds (Sassen and Cho, 1992;Szczap et al., 2000;Carlin et al., 2002;Lynch et al., 2002), as well as the values of D eff .Figure 1a shows an example of a 10 km × 10 km optical thickness field at 12.05 µm with a spatial resolution of 100 m and Fig. 1b  System) FSSP 300 probes, more sensitive to large particles.The extinction coefficient is obtained by the Polar Nephelometer (PN) and the cirrus mean optical thickness by IIR measurements.In addition, the meteorological profiles (wind speed and orientation, temperature, humidity etc.) are set in the model (Fig. 2) using the meteorological data provided by the European Center for Medium-Range Weather Forecasts (ECMWF) with adaptations of the potential temperature and relative humidity profiles necessary to form cirrus (Starr and Cox, 1985).
The scale invariant properties are controlled by a −5/3 constant spectral slope at all the scales and altitude levels according to the cirrus backscattering coefficient at 532 nm measured at different altitudes by the LIDAR CALIOP/CALIPSO (Fig. 3), and the extinction coefficient measured by the Polar Nephelometer at the aircraft altitude.
To compare real measurements and 3DCloud simulations, the MODIS "true color RGB" picture of the cirrus is presented in Fig. 4a. Figure 4b corresponds to the CALIOP/CALIPSO vertical profile of the cirrus attenuated backscattering coefficient.Figure 4c represents the 20 km × 20 km optical thickness field at a spatial resolution of 100 m generated by 3DCloud inside the black rectangle of Fig. 4a and Fig. 4d the IWC profile with a vertical resolution of 58 m.On Fig. 4a and Fig. 4c, the lines in the cirrus have the same orientation.It illustrates that the cirrus generated by 3DCloud and that one observed on 25 May 2007, during the CIRCLE II campaign, have a similar geometry.The mean optical thickness is set τ c = 0.41 in the model as that retrieved from the IIR measurements in the black rectangle area.Furthermore, the comparison of Fig. 4b and 4d allows us to see that the vertical profiles of the simulated and observed cirrus present the same cloud top and base altitudes.Three different simulations were done from the cirrus field observed during CIRCLE II and their properties are summarized in Table 1.The cirrus CII-1 corresponds to the simulation of the cirrus observed on 25 May.The cirrus CII-2 is the same with the IWC increased twofold.The cirrus CII-3 has the same distribution of IWC but with optical properties of the cirrus 8. Their properties are resumed in    Full

Cirrus optical property parametrization
Cirrus microphysical and optical properties are particularly difficult to apprehend because of the variability of shapes, sizes and orientations of ice crystals that can exist.Numerous studies have treated this problem and used different methods to compute the optical properties of cirrus clouds for visible and infrared wavelengths (Magono, 1966;Labonnote et al., 2000;Yang et al., 2001Yang et al., , 2005;;Baum et al., 2005Baum et al., , 2011; Baran and Labonnote, 2007;Baran et al., 2009Baran et al., , 2011a, b), b).Among all the available methods, we choose for the cirrus 1 to 8 to use the ice crystals model developed by Yang et al. (2001Yang et al. ( , 2005)).This model allows us to supply an extinction coefficient, a single scattering albedo and an asymmetry factor for seven forms of crystals having an effective diameter from 1 µm to 10 000 µm.In our simulations, the aggregate shape with a monodisperse distribution is selected because it is one of those used in the IIR retrieval algorithm (Garnier et al., 2013).Furthermore, in the thermal infrared, the forward peak is weak and the particle phase functions are smooth enough to be approximate in a correct way (Yang et al., 2001) by phase functions of Henyey-Greenstein type (Henyey and Greenstein, 1940) that is assumed in the Yang et al. (2001Yang et al. ( , 2005) ) model.Values of the extinction coefficient efficiency, the single scattering albedo and the asymmetry parameter are presented in Table 2.
For the CII-1 and CII-2, the parametrization developed by Baran et al. (2009Baran et al. ( , 2011a, b) , b) is used to study the impact of the optical property variabilities on the TOA BT.This parametrization consists in obtaining the extinction coefficient σ e , the single scattering albedo 0 and the asymmetry factor g from the couple (IWC, Temperature).The relations between the optical properties and the couple (IWC, Temperature) were obtained from more than 20 000 Particle Size Distributions (PSD) provided by in situ measurements (Field et al., 2005(Field et al., , 2007)).Therefore, from a realistic 3-D profile of IWC and temperature, we employed this parametrization to generate the 3-D heterogeneous optical property field for the CII-1 and CII-2 cirrus.Their vertical optical property distributions are presented in Fig. 5. Introduction

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Radiative transfer modelling
Thermal radiative transfer computations are made with the 3-D Monte-Carlo code, 3DMCPOL, initially developed in the visible range by Cornet et al. (2010) and extended for this study to the Thermal InfraRed (TIR).In 3DMCPOL, the atmosphere is divided into voxels (3-D pixels), with a constant horizontal size (dx, dy) and a variable vertical size dz.Each of the voxels is described by the cloud optical properties: the extinction coefficient σ e , the single scattering albedo 0 , the phase function and the cloud temperature T c .3DMCPOL is a forward Monte-Carlo which used the Local Estimate Method (LEM, Marshak and Davis, 2005;Mayer, 2009).It was first developed for the visible range where the initial direction of photon packages is given by the solar direction.The extension of the code to the TIR conserved the forward method, even if the source of emission can be the atmosphere, the surface or the cloud.Monte-Carlo methods consist in following photon packages, which undergo some scattering and absorption processes in the atmosphere and surface reflections.At each scattering event, the LEM is used and consists to compute the contribution of emission, scattering or reflection events into the detector direction, attenuated by the medium optical thickness crossed between the interaction and the detector.The LEM weight W le attached to the photon is thus defined as: with P 11 the first element of the scattering matrix which gives the probability of a photon to be scattered in the direction of the detector, θ s the scattering angle between photon direction and the detector, τ m the medium optical thickness from the interaction to the detector and θ v the view zenith angle.W 0 is the weight due to the cloud absorption and it corresponds at each interaction to the product of the single scattering albedo 0 for cloud scattering (or of the surface albedo α for surface reflection).When W 0 is less Introduction

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Full than 10 −6 , the contribution of the photon package is considered to be neglectful and a new one is launched.
To include the gaseous absorption in 3DMCPOL, we use the correlated k-distribution method (Lacis and Oinas, 1991;Kratz, 1995).This technique allows us to take into account the absorption variations in each spectral band by dividing the range of absorption coefficient values into bins using a sum of weighted exponentials.To each bin, a weight a ib is associated whose sum is equal to 1.However, to avoid running radiative transfer for every bin which is very time consuming, we use the Equivalence Theorem (Partain et al., 2000;Emde et al., 2011).It consists in attaching an absorption vector W g to the photon package, with a size corresponding to the bin number n bin of the correlated k -distribution.W g (ib) is expressed by Eq. (2).
with ib the number of correlated k-distribution bin, l the photon package path and k g (ib, z) the absorption coefficient for bin ib and altitude z.
With the gaseous absorption, the LEM weight W le becomes a vector of size n bin as: With this method, it is possible to compute the reflectance for a spectral band with variable gaseous absorption with one Monte-Carlo RT simulation, as long as the medium optical properties are homogeneous.The other important modification concerns the sources of emission.The total number of emission processes (or photon packages) is fixed in input by the user and a fraction of the total number of photon packages corresponds to each source (cloud, surface and gases).This fraction is obtained in function of the emission characteristics (emissivity, temperature) of all the corresponding voxels.We compute a quantity called source flux F , defined as the energy Introduction

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Full emitted in every direction by each cell of the emission source.As emission processes are isotropic, the direction of emission is chosen randomly.The sum of the source flux of the cloud, the surface and the gases (F c +F s +F g respectively) over all the cells gives the total emitted energy.The number of photons emitted by a source type is then the proportion of energy of this source to the total energy.A random choice determines the spatial location where the emission takes place.
The source flux F is computed from the radiance R. For cloud cell, the source flux F c is defined as: with R c the cloud radiance emitted by the cell, τ a and τ e the absorption and extinction optical thicknesses respectively, B(T c ) the Planck function at the temperature T c and 0 the single scattering albedo.In the same way, the source flux F s emitted from the surface is defined as: with R s , s and T s the surface radiance, the emissivity and the surface temperature respectively.Finally, the source flux F g emitted by the gases in function of bin ib is expressed by: with a i , the weight associated to bin ib, τ g (ib) the gaseous optical thickness.
Validation of the 3DMCPOL modifications was done by inter-comparisons with SHDOM simulations (Evans, 1998).An example of comparison between 3DMCPOL and SHDOM is illustrated in Fig. 6 at the scale of 100 m × 100 m, for the cirrus 3 and for radiances along the X-axis for Y = 5 km, Fig. 6c shows the relative difference (%) field between 3DMCPOL and SHDOM radiances and Fig. 6d represents the correlation plot between 3DMCPOL and SHDOM radiances.The relative error between 3DMCPOL and SHDOM simulations is generally under 2 % and the correlation between the two models is 0.998.The small remaining differences can be explained by the different treatment of the medium properties as 3DMCPOL considers the medium properties homogeneous in each voxel while SHDOM interpolates the properties in a voxel.Comparisons were also been made for several cases of cirrus and spectral bands and give similar results.
3 Heterogeneity effects on the brightness temperatures simulated at TOA

Description of the heterogeneity effects
Clouds present many variabilities at different scales.In retrieval algorithms, for simplifications and computational reasons, the Independent Column Approximation (ICA, Stephens et al., 1991) is commonly applied: the cloud layers are assumed to be vertically and horizontally homogeneous and independent of each other.At the scale of IIR (or MODIS) observation pixel (1 km × 1 km), the radiative transfer is supposed to be 1-D without horizontal transport between columns.Cloud properties are assumed to be homogeneous in each cloudy layer.To study the cirrus heterogeneity effects that can affect the BT observed in the thermal infrared at TOA, we made 3-D and 1-D simulations with 3DMCPOL.3-D BT are simulated at the spatial resolution of 100 m × 100 m and then averaged to 1 km × 1 km (BT3D 1 km ).For 1-D BT, cloud properties are first averaged at 1 km × 1 km before simulating BT (BT1D 1 km ).Note that the spatial resolution is an important parameter in the study of the impact of cirrus heterogeneities on BT and our results are, thus, only applicable for a spatial resolution close to 1 km × 1 km.
To describe the heterogeneity effects, we plot in Fig. 7  the three IIR channels (8.65 µm, 10.60 µm and 12.05 µm).We see that the relation is non-linear and the averaging of BT leads to Jensen inequality, usually called the Plan Parallel Approximation (PPA).The width of the BT3D 100 m distribution, which is about 4-5 K, is due, on the one hand, to the vertical variability of the extinction coefficient and, on the other hand, to the photon horizontal transport between cloud columns (Varnai and Marshak, 2001).The PPA thus causes the average of BT (BT3D) to be higher than the BT of the average of the optical thicknesses τ.In Figs. 8 and 9, we plot the absolute value of the BT differences at the scale of 100 m (|∆BT 100 m |) and 1 km (|∆BT 1 km |) respectively for a case of a very inhomogeneous cirrus (cirrus 5).Note that, for 3D brightness temperatures at 100 m × 100 m (BT3D 100 m ), the extinction coefficient varies vertically, and not for the 1-D brightness temperatures at 100 m × 100 m (BT1D 100 m ).Several effects contribute to the differences between 3-D and 1-D BT and are described below for the 100 m spatial resolution (Fig. 8): -In green, we plot the absolute value of the extinction vertical heterogeneity effect |BT1Dvhe 100 m −BT1D 100 m |, with BT1Dvhe 100 m corresponding to 1-D radiative transfer with independent column and vertically heterogeneous extinction coefficient.
-In red, we plot the absolute value of the BT difference due to the horizontal transport which is computed with the calculation of |BT3Dvho 100 m − BT1D 100 m | with BT3Dvho 100 m corresponding to 3-D radiative transfer with vertically homogeneous extinction.
-In blue, we plot the absolute value of the difference |∆BT 100 m | obtained with the calculation of |BT3D 100 m − BT1D 100 m |.Note that |∆BT 100 m | is not the sum of the two effects described above, because some of them can be opposing.
In addition, we also plot the absolute value of the statistical error (black line) due to the statistical approach of the Monte-Carlo algorithm which is about 0.5 K. Introduction

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Full We see that at 100 m, the |∆BT 100 m | for the band at 8.65 µm is larger and dominated by the horizontal transport effect because the scattering is higher for this band than the two others.On the contrary, the vertical extinction variability, which is also associated to the vertical emissivity variability, is larger at 10.60 µm and at 12.05 µm than for the band at 8.65 µm.For these two bands, the horizontal transport effect and the vertical extinction variability are of the same order and thus, contribute with equal prominence to the total differences |∆BT 100 m |.The results are similar for the other cirrus cases and they are not presented here.
At the scale of 1 km × 1 km (Fig. 9) the horizontal transport (red) and the extinction vertical heterogeneity (green) effects are slight.That means that the absolute difference |∆BT 1 km | (blue) between BT3D 1 km and BT1D 1 km are strongly dominated by the PPA.It reaches more than 12 K for the largest optical thicknesses in the three bands.Horizontal transport in the band at 8.65 µm is larger than in the other bands, but remains low compared to the total difference |∆BT 1 km |.The extinction vertical inhomogeneity appears also negligible at the 1 km×1 km scale.In summary, Fig. 9 shows that, at the scale of IIR and MODIS, the most important difference between BT3D 1 km and BT1D 1 km is due to the PPA.During the retrieval process, the cloud properties retrieved from brightness temperatures at the 1 km × 1 km pixel will thus be different from the mean cloud properties.

Heterogeneity effects due to optical thickness variabilities
This section focuses on the impact of the optical thickness variabilities on TOA BT, all the other parameters being constant (atmospheric profile, altitude, geometrical thickness, surface temperature, particles size and shape).Figure 10 presents the heterogeneity effects on the brightness temperatures (∆BT 1 km ) for the band at 12.05 µm in function of the optical thickness τ 1 km (Fig. 10a), the standard deviation of the optical thickness σ τ 1 km computed from the 100 pixels of 100 m × 100 m (Fig. 10b), and the heterogeneity parameter ρ τ 1 km at 1 km × 1 km (Fig. 10c).We see that the difference ∆BT 1 km is better correlated with σ τ 1 km (R = 0.95).Indeed, as discussed previously, the 27471 Introduction

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Full PPA is the most important effect at this scale and the larger σ τ 1 km is, the greater the averaging effect is and, thus, the difference ∆BT 1 km .Furthermore, as the increase in the τ 1 km is not totally correlated with the increase in the sub-pixel heterogeneity σ τ 1 km , the correlation between ∆BT 1 km and τ 1 km is smaller, as well as the correlation between ∆BT 1 km and ρ τ 1 km .The σ τ 1 km appears, therefore, to be the best parameter to highlight the heterogeneity effects and it will always be used afterwards to represent the field heterogeneity.
The ∆BT 1 km in function of the standard deviation σ τ 1 km for the different cirrus of Table 1 and for the three channels of the IIR are presented in Fig. 11.The first remark on this figure is that correlation coefficient R is better than 0.80 for all the cirrus, except for cirrus 1.For this cirrus, the lowest optical thickness (0.45) associated to the strong scattering at 8.65 µm lead to smoothing of the variability due to the photon transport.The five cirrus have similar characteristics with different mean optical thickness τ c and heterogeneity parameter ρ τ .For each of these cirrus, we computed BT3D 1 km and BT1D 1 km for 100 pixels of 1 km × 1 km.We see that the relation between ∆BT 1 km and σ τ 1 km is almost linear.This figure shows that, for the same altitude, geometrical thickness and optical properties, ∆BT 1 km depends mainly on the optical thickness subpixel heterogeneity, the optical thickness sub-pixel distribution being almost insignificant.Note that, similar results were obtained in the visible (Szczap et al., 2000;Cornet et al., 2004) where heterogeneity effects depend mainly on τ 1 km and σ τ 1 km .
We now study the impact of heterogeneities for different ice crystal effective diameters D eff .In Fig. 12, we present ∆BT 1 km in function of σ τ 1 km for D eff = 9.95 µm, 20.09 µm and 40.58 µm.The crystal model used is the P. Yang model (Yang et al., 2001(Yang et al., , 2005, Table 2), for an aggregate shape.This figure shows that ∆BT 1 km decrease with the increase in D eff for the three TIR bands.For the bands at 10.60 µm (Fig. 12b) and at 12.05 µm (Fig. 12c), the single scattering albedo 0 and the asymmetry parameter g increase with D eff (Table 2), that lead to a decrease in the absorption inside each 1 km×1 km pixel and an increase in energy in the forward peak respectively.Therefore, photons emitted from the surface cross easier the cloud, leading to a decrease in the Introduction

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Full These two effects are opposite and the ∆BT 1 km do not change significantly between these two effective sizes.In summary, the amplitude of heterogeneities effect depends on the particle effective size when a constant size is supposed in the cloud.Indeed, in this situation, the optical properties can be very different from one crystal size to another and, thus, lead to different values.For example, the mean value of ∆BT 1 km for the cirrus 3 is 3.12 K (band at 12.05 µm), while for cirrus 7 the mean value is 0.99 K.However, a unique crystal size and shape as supposed in these simulations is not realistic.It just allows us to give information on the crystal size influence on TOA BT.In reality, a large variety of particle sizes and shapes are present in a cirrus and, therefore, a more complex parametrization of the optical properties is necessary to study the impact of the optical property variabilities on TOA BT.

Heterogeneity effects due to optical and microphysical property variabilities
To study the effects of three dimensional optical property heterogeneities on TOA BT and simulate more realsitic clouds, we employed the parametrization developed by Baran et al. (2009Baran et al. ( , 2011a, b, b) (see Sect. 2.2).The cirrus simulated from the CIRCLE II campaign are used to generate a heterogeneous macrophysical field (Fig. 4 and Table 1) and optical property field (Fig. 5).
Before studying the heterogeneities' effect, we look at the effects of vertical variabilities of the extinction coefficient σ e , the single scattering albedo 0 and the asymmetry factor g (Fig. 13).The 1-D RT with homogeneous columns (BT1D 1 km ) is compared with the 1-D RT with heterogeneous columns (BT1Dhe 1 km ) for the cirrus CII-2.We see that the difference is, on average, lower than 0.5 K for the three channels and it is maximum 27473 Introduction

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Full for the band at 8.65 µm.Indeed, because the scattering is higher in this band and thus the absorption lower, the vertical optical property variabilities have, therefore, a more significant impact on BT.
∆BT 1 km in function of σ τ 1 km are presented in Fig. 14 in the three IIR channels for cirrus CII-1 and CII-2 with variable optical properties and for cirrus CII-3 with homogeneous optical properties.This last cirrus has been generated in order to compare, on the one hand, the two optical property models and, on the other hand, the influence of the cirrus vertical extension (see Sect. 3.4).In this figure, the mean correlation coefficient R averaged on the three channels between ∆BT 1 km and σ τ 1 km is larger again than 0.80.The low value of R (0.88) for cirrus CII-1 is due to the band at 8.65 µm.Indeed, in this case, the PPA bias is close to zero, firstly because the σ τ 1 km values are quite small the cirrus CII-1 and, secondly, because there is more scattering which tends to smooth the field heterogeneity and, thus, decorrelates the relation between ∆BT 1 km and σ τ 1 km .For this cirrus, which is the result of the realistic simulation obtained from the cirrus observed during the CIRCLE II campaign, the heterogeneities' effect on the TOA BT are, on average, lower than the instrumental precision of the IIR (1 K).The PPA and IPA bias have, thus, a limited impact in this case.Concerning the cirrus CII-3 with homogeneous optical properties, the ∆BT 1 km is lower than for other cirrus for the band at 8.65 µm, but this difference decreases for the band at 10.60 µm and becomes positive for the band at 12.05 µm.Indeed, cirrus CII-3 contains only small aggregate crystals (D eff = 9.95 µm).As explained in Sect.3.2, the PPA is larger for the band at 12.05 µm because small particles are highly absorbing in this band (Table 2).For the band at 8.65 µm, 0 is more important for small particles leading to more scattering which, thus, smooth the heterogeneities.We can, thus, conclude that the optical property model has a weak influence on the heterogeneity effects on BT (in average inferior to 1 K).Introduction

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Influence of altitude and geometrical thickness of the cirrus cloud
In the troposphere, the temperature decreases with altitude.A cirrus with a high altitude emits, therefore, at a lower temperature than a cirrus with the same optical properties at a lower altitude.Figure 15 presents ∆BT 1 km in function of σ τ 1 km for cirrus 2 with the top altitude of 7.97 km, the cirrus 8 and CII-3 with the top altitude of 11.06 km.We note that ∆BT 1 km is greater for the higher cirrus.Indeed, for a higher cloud, the BT contrast between the cloud top and the clear sky is larger and, thus, the PPA bias is also larger.
To study the influence of the vertical extension, we then compare the cirrus 8 and CII-3.They have the same optical characteristics, the same cloud top altitude but a different vertical extension involving a different cloud base at 9.06 km for cirrus CII-3 and at 10.60 km for cirrus 8.Note that, the cirrus optical thickness distribution is also different but, as seen previously, it does not influence the ∆BT 1 km much.Figure 15 shows slight differences between the two clouds at 8.65 µm and at 10.60 µm but a greater difference at 12.65 µm.At this wavelength, the absorption is strong, leading to a large contrast between the cloud top and clear sky brightness temperatures.However, the cloud base of cirrus CII-3 is at a lower altitude, meaning that the emission temperature is higher and, thus, closer to the surface temperature.The average temperature of cirrus CII-3 is, thus, less contrasted with the surface temperature than for cirrus 8, leading to a lower PPA bias.To conclude, for two identical cloud top altitudes, if the band is strongly dominated by the absorption (as the band at 12.05 µm), the PPA bias decreases with the increase in the vertical extension, but this effect is, on average, in the order of tenths of a Kelvin.

Influence of the observation geometry on cloud heterogeneity effects
The previous results were presented for nadir view as measured by IIR/CALIPSO.In this section, heterogeneity effects for other view directions are investigated.In the framework of a 3-D radiative transfer for tilted geometry, photons cross different inhomogeneous columns (TIPA, Várnai and Davies, 1999)  which assumes infinite layers.Figure 16 shows the ∆BT 1 km averaged over all the pixels of the field (∆BT 1 km ) in function of the zenith view angle Θ v for cirrus 1 to 5 and for CII-1, CII-2 and CII-3 cirrus.For the band at 8.65 µm, the ∆BT 1 km increases with Θ v .For cirrus 1 to 5, its value is twice as high for Θ v = 60 • than for Θ v = 0 • .For CII-1, CII-2 and CII-3 cirrus, the difference is approximately 10 times as large because they present more important 3-D extinction coefficient variability.In the band at 12.05 µm, we note for cirrus 3, cirrus 4 and cirrus 5 that the ∆BT 1 km does not change in function of Θ v .This is due to a saturation effect.These three cirrus have a large mean optical thickness (τ c = 1.8), with optical thicknesses superior to 6 for some pixels.As we can see in Fig. 7, the PPA tends to zero for these pixels as the surface emission is not visible any more.We also note that, for Θ v = 0 • , ∆BT 1 km is no longer correlated with σ τ 1 km which is computed for integrated vertical cloud column and, thus, does not represent the heterogeneity perceived from an oblique view zenith angle.Concerning view azimuth angle Φ v , we see no real tendency, except differences due to different crossed optical properties.

Summary and conclusions
This paper presents results concerning the impact of cirrus cloud heterogeneities on the top-of-atmosphere brightness temperatures (∆BT 1 km ).Spatial radiometers such as IIR/CALIPSO and MODIS/AQUA measure top-of-atmosphere brightness temperatures at 1 km × 1 km and their operational algorithms use the Independent Pixel Approximation (IPA) and the Plan-Parallel Approximation (PPA) to retrieve cirrus cloud properties.Different effects can result from this assumption.We have shown that the larger effect at the 1 km scale is the PPA.It depends on the absorption properties and therefore on the observation channel with larger effects for bands at 10.60 µm and 12.05 µm.We have also shown that ∆BT 1 km depend mainly on the optical thickness standard devia-Introduction

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Full tion inside the observation pixel and on the brightness temperature contrast between the cloud top and the clear sky atmosphere.
To summarize the different results presented in this paper, Fig. 17 shows the brightness temperature differences ∆BT 1 km simulated to the nadir in function of the optical thickness at 1 km × 1 km (τ 1 km ).The two red lines correspond to the instrumental precision of IIR.The different cases of cirrus are superimposed to obtain 2, 000 pixels with different optical thicknesses.For these cirrus, the contrast between the cloud top and clear sky brightness temperatures are between −46 K to −67 K and we observe that the heterogeneity effects start to be significant (∆BT 1 km higher than the IIR instrumental precision of 1 K) around τ 1 km ∼ 0.4 at 12.05 µm.This is equivalent to τ 1 km ∼ 0.3 at 532 nm, corresponding to the limit of optically thick cirrus defined by Sassen and Cho (1992).In term of σ τ 1 km , heterogeneity effects are superior to 1 K for σ τ 1 km ∼ 1.
Furthermore, we have also shown that the impact of heterogeneity effects increases strongly with the zenith view angle except for cirrus with large mean optical thickness (τ c = 1.8),where a saturation effect is observed for the band at 12.05 µm due to the strong absorption.
The next step will be the estimation of the impact of cirrus heterogeneities on the cloud properties retrieved from IIR thermal infrared measurements with the 1-D approximation (Fauchez et al., 2013).Introduction

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Full Table 1.Cirrus simulated by 3DCloud."OP" corresponds to the optical properties parametrization, "CTA" corresponds to the Cirrus Top Altitude, "Yal" represents the model of ice crystals developed by Yang et al. (2001Yang et al. ( , 2005) ) for aggregates ice crystals and "Bal" represents the parametrization of ice crystals optical properties developed by Baran et al. (2009Baran et al. ( , 2011a, b), b).Full  Full         The two red lines correspond to the instrumental precision of the IIR of about ± 1 K. 28 Fig. 17.Brightness temperature differences ∆BT 1 km view at the nadir between BT3D 1 km and BT1D 1 km in function of the optical thickness at 1 km (τ 1 km ) for the three channels of the IIR and for 2000 pixels per band.The two red lines correspond to the instrumental precision of the IIR of about ±1 K.

Cirrus
Discussion Paper | Discussion Paper | Discussion Paper | used to simulate the radiative transfer inside Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | the x-z view of the Ice Water Content (IWC).The second cirrus is generated from measurements obtained on 25 May 2007 during the CIRCLE II campaign (Mioche et al., 2010).CIRCLE II was an airborne campaign dedicated to the study of the cirrus optical properties and the validation of space measurements made by the LIDAR CALIOP and the Infrared Imaging Radiometer IIR on board CALIPSO.This campaign consists of two Falcon 20 aircraft with several complementary on-board instruments to study cirrus cloud.In order to simulate the cirrus observed during this campaign in a realistic way, we used in input of 3DCloud the in situ measurements provided by the aircraft as well as IIR and MODIS radiometric measurements.The cirrus mean IWC is determined by the combination of the CPI (Cloud Particle Imager), more sensitive to small particles, and the PMS (Particle Measuring Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Three and half billion photon packages are launched, which lead to a statistical accuracy of about 10 −3 W m −2 sr −1 .Figure6ashows the 3DMCPOL radiance field view at the nadir, Fig.6bpresents the comparison of 3DMCPOL and SHDOM Discussion Paper | Discussion Paper | Discussion Paper | the variation of the BT at 100 m (BT3D 100 m ) in function of the optical thickness at 100 m (τ 100 m ) for the cirrus 5 and for Discussion Paper | Discussion Paper | Discussion Paper | Screen / Esc Printer-friendly Version Interactive Discussion Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Screen / Esc Printer-friendly Version Interactive Discussion Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |contrast between high and low optical thicknesses.The radiative field heterogeneities appear thus smoothed.Concerning the band at 8.65 µm, 0 decreases with D eff but the asymmetry parameter g increases.Between D eff = 9.95 µm and D eff = 20.09µm, the increase in g dominates and, as previously explained, the ∆BT 1 km decrease.But between D eff = 20.09µm and D eff = 40.58µm, g increases slightly and 0 decreases.

Fig. 1 .
Fig. 1.(a) 10 km × 10 km optical thickness field at 12.05 µm with a horizontal spatial resolution of 100 m, (b) x-z view of the cirrus Ice Water Content (IWC) with a vertical spatial resolution of 58 m.

Fig. 1 .Fig. 2 .Fig. 2 .
Fig. 1.(a) 10 km × 10 km optical thickness field at 12.05 µm with a horizontal spatial resolution of 100 m, (b) x-z view of the cirrus Ice Water Content (IWC) with a vertical spatial resolution of 58 m.

Fig. 3 .
Fig. 3. Spectral slopes of the CALIOP/CALIPSO backscattering coefficient at 532 nm observed at 10.955 km (a), 10.356 km (b) and 9.900 km (c) in function of the frequency in Hz