Impact of cirrus clouds heterogeneities on top-of-atmosphere thermal infrared radiation

This paper presents a study of the impact of cirrus cloud heterogeneities on the thermal infrared brightness temperatures at the top of the atmosphere (TOA). Realistic 3-D cirri are generated by a cloud generator based on simplified thermodynamic and dynamic equations and on the control of invariant scale properties. The 3-D thermal infrared radiative transfer is simulated with a Monte Carlo model for three typical spectral bands in the infrared atmospheric window. Comparisons of TOA brightness temperatures resulting from 1-D and 3-D radiative transfer show significant differences for optically thick cirrus ( τ > 0.3 at 532 nm) and are mainly due to the plane-parallel approximation (PPA). At the spatial resolution of 1 km× 1 km, two principal parameters control the heterogeneity effects on brightness temperatures: i) the optical thickness standard deviation inside the observation pixel, ii) the brightness temperature contrast between the top of the cirrus and the clear-sky atmosphere. Furthermore, we show that the difference between 1-D and 3-D brightness temperatures increases with the zenith view angle from two to ten times between 0 ◦ and 60 due to the tilted independent pixel approximation (TIPA).


Introduction
Cirrus clouds cover 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 and atmosphere.By contrast, a part of the solar incident radiation is reflected to space because of the parasol effect, but this is generally slight for high clouds.
Therefore, cirrus clouds lead to a positive radiative effect (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).
Global observations are well adapted to follow and better understand cloud evolution and characteristics.With this aim, many satellites are dedicated to cloud obervations from visible to microwave ranges.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 biases in the retrieval of cloud properties.
In this context, radiative transfer modeling is very useful for studying the 1-D bias as a 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 fluxes or heating/cooling rates.Concerning fluxes, Hogan and Kew (2005) showed that radiative transfer calculations using IPA can change the mean The mean Top Of Atmosphere (TOA) radiative fluxes by about 45 W m −2 in the shortwave and 15 W m −2 in the long wave.Furthermore, Chen and Liou (2006) showed that significant impact exists on the broadband thermal cooling rates (around 10 %) Published by Copernicus Publications on behalf of the European Geosciences Union.
T. Fauchez et al.: Cirrus heterogeneities in the thermal infrared 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 (BTs) or radiances 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., 2014), used to generate realistic cloud scenes, and the Monte Carlo 3-D radiative transfer code named 3DMCPOL (Cornet et al., 2010), used to simulate the radiative transfer inside three dimensional (3-D) atmospheres.We simulate BTs for several cirri generated from realistic conditions as well as from measurements made during the CIRrus CLoud Experiment-II (CIRCLE II) airborne campaign (Mioche et al., 2010;Sourdeval et al., 2012).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 cirri 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., 2014).First, basic atmospheric equations with idealized meteorological profiles are resolved in order to simulate the 3-D IWC.Then, 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 spectrum 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 preferably at smaller scales.However, as we show later with lidar measurements, we do not observe Table 1.Cirri 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 crystal optical properties developed by Baran et al. (2009); Baran (2012); Baran et al. (2013) a change in the spectral slope with the altitude (see 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 realization, 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 being the standard deviation of the optical thickness.Eight cirri 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 typical 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 the x-z view of the IWC.
The second cirrus is generated from measurements obtained on the 25 May 2007 during the CIRCLE II campaign (Mioche et al., 2010).CIRCLE II was an airborne campaign dedicated to the study of cirrus optical properties and the validation of space measurements made by the Cloud-Aerosol Lidar with Orthogonal Polarization (CALIOP; Winker et al., 2009) and the Infrared Imaging Radiometer (IIR; Garnier et al. (2012Garnier et al. ( , 2013) )   campaign consists of two Falcon 20 aircraft with several complementary onboard instruments to study cirrus cloud.In order to simulate the cirri observed during this campaign in a realistic way, we used the in situ measurements provided by the aircraft as well as IIR and MODIS (Moderate-Resolution Imaging Spectroradiometer;Platnick et al., 2003) radiometric measurements as input of 3DCloud.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 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 cirri (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 Red Green Blue (RGB)" picture of the cirrus is presented in Fig. 4a. Figure 4b corresponds  (Yang et al., 2001(Yang et al., , 2005) )   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 4d represents the IWC profile with a vertical resolution of 58 m.In Fig. 4a and c, 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 to τ c = 0.41 in the model as is that retrieved from the IIR measurements in the black rectangle area.Furthermore, the comparison of Fig. 4b and d 1.

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;C. 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., 2009;Baran, 2012;Baran et al., 2013).Among all the available methods, we choose for 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 approximated by the Henyey-Greenstein phase function (Henyey and Greenstein, 1940) and this 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.The model of Yang et al. (2001Yang et al. ( , 2005) ) based on the effective diameter does not allows us to represent its threedimensional variability in the cirrus.To study the impact of the optical property variabilities on the TOA BT, we use the parametrization developed by Baran (2012) and Baran et al. (2013) for the CII-1 and CII-2 cirrus.Baran et al. (2009) show that the extinction coefficient σ e , the single scattering albedo 0 and the asymmetry factor g could be related to the couple (IWC, Temperature).The relations between 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.The corresponding vertical temperature profile is shown in Fig. 2 and the IWC vertical profile in Fig. 4d.

Radiative transfer modeling
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 at-mosphere 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 of computing the contribution of emission, scattering or reflection events into the detector direction, attenuated by the medium optical thickness between the place of interaction and the detector.
The LEM weight W le attached to the photon is thus defined as with P 11 being the first element of the scattering matrix which gives the probability of a photon being 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 zenith view 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 than 10 −6 , the contribution of the photon package is considered to be negligible 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 assigned 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 of attaching an absorption vector W g to the photon package, with a size corresponding to the bin number nbin of the correlated k-distribution.W g (ib) is expressed by Eq. ( 2).
with ib being the number of the 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 nbin as follows: With this method, it is possible to compute the reflectance for a spectral band with variable gaseous absorption with one Monte Carlo radiative transfer (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 the 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 as a function of the emission characteristics (emissivity, temperature) of all the corresponding voxels.We compute a quantity called the source flux F , defined as the energy 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 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 being the surface radiance, the emissivity and the surface temperature, respectively.Finally, the source flux F g emitted by the gases as a function of bin ib is expressed by with a i being the weight associated to bin ib and τ g (ib) the gaseous optical thickness.
Validation of the 3DMCPOL modifications was done by intercomparisons with Spherical Harmonic Discrete Ordinate Method (SHDOM) simulations (Evans, 1998).An example of a comparison between 3DMCPOL and SHDOM is illustrated in Fig. 6 at the scale of 100 m × 100 m for cirrus 3 and for θ v = φ v = 0 • .Three and half billion photon packages were launched, which lead to a statistical accuracy of about 10 −3 W.m −2 .sr−1 .Figure 6a shows the 3DM-CPOL radiance field of view at the nadir, Fig. 6b presents the comparison of 3DMCPOL and SHDOM 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 3DM-CPOL 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, with a different geometry of observation (θ v = 30 • ,60 • and φ v = 45 • , 90 • , 180 • ) and spectral bands (at 8.65 µm and 12.05 µm), and gave 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 simplification and for computational reasons, the independent column approximation (ICA; Stephens et al., 1991) is commonly applied: cloud layers are assumed to be vertically and horizontally homogeneous, independent of each other with an infinite horizontal extent.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.
To study cirrus heterogeneities that can affect 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 (BT 3D 1 km ).For 1-D BT, cloud properties are first averaged at 1 km × 1 km before simulating BT (BT1-D 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, in Fig. 7 we plot the BT at 100 m (BT3D 100 m ) as a function of the optical thickness at 100 m (τ 100 m ) for cirrus 5 and for the three IIR channels (8.65 µm, 10.60 µm and 12.05 µm).We see that the relation is nonlinear and the averaging of BT leads to Jensen inequality, usually called the plane-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 optical thicknesses τ .
In Figs. 8 and 10, 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).We plot for brightness temperatures, but results are similar in radiance space.Note that, for 3-D brightness temperatures at 100 m × 100 m (BT3D 100 m ), the extinction coefficient varies vertically, but not for the 1-D brightness temperatures at 100 m × 100 m (BT1-D 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 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.
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 greater for this band than for the two others.By contrast, 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.
To evaluate the horizontal photon transport effects on TOA BT, we present a step cirrus cloud in Fig. 9.The optical thickness described at the scale of 100 m (τ 100 m ) is equal to 3.5 between 0 km and 2 km and to 0 after that."Thin cirrus" (solid lines) and "thick cirrus" (dashed lines) are two cirri with the same optical thickness and the same top altitude but with different geometrical thickness (0.4 km and 2 km, respectively).Curves represent the 3-D brightness temperatures whereas the straight lines correspond to 1-D brightness temperatures computed using the IPA assumption.Logically, pixels with optical thickness equal to zero have large BTs corresponding to clear-sky atmosphere, while pixels with larger optical thickness have lower BTs corresponding to the top of the cirrus.Due to the stronger absorption at 12.05 µm, the BT differences between opaque pixels and clear-sky pixels are greater for the band at 12.05 µm than at 8.65 µm.We note also that BTs of "thick cirri" are larger than those of "thin cirri" because for the "thick cirrus", the energy emitted by the cloud base, which is closer to the surface, is greater.Photon horizontal transport effects are visible near the optical thickness transition (between τ 100 m = 0 and 3.5), where photons emitted from the surface cross over or are scattered in the cloud leading to an increase in the BT.This increase goes  further in the case of the "thick cirrus" because the mean extinction coefficient per cell is smaller and photons can spread further.For bands at 10.60 µm and at 12.05 µm, with larger absorption, cloud pixels are impacted until 100 m for "thin cirri "and 400-500 m for "thick cirri".At 8.65 µm, cloud pixels up to 1 km from the cloud edge can be impacted.We note that, based on the IIR accuracy of 1 K, photon horizontal transport effects become, on average, larger than this accuracy below a spatial resolution of 250 m.At the scale of 1 km × 1 km (Fig. 10) the horizontal transport (red) and the extinction vertical heterogeneity (green) effects are slight.This means that the absolute difference | BT 1 km | (blue) between BT3D 1 km and BT1 − D 1 km is strongly dominated by the PPA.It reaches more than 12 K for the largest optical thicknesses in the three bands.Hor-izontal 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 also appears to be negligible at the 1 km × 1 km scale.In summary, Fig. 10 shows that, at the scale of IIR and MODIS, the most important difference between BT3D 1 km and BT1 − D 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.Figure 11.Brightness temperature differences, BT 1 km , as a function of the optical thickness τ 1 km , the optical thickness standard deviation σ τ 1 km and the heterogeneity parameter ρ τ 1 km for different cirri presented Table 1 and for the band at 12.05 µm.R represents the correlation coefficient computed for the five cirri.

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 11 presents the heterogeneity effects on the brightness temperatures ( BT 1 km ) for the band at 12.05 µm as a function of the optical thickness τ 1 km (Fig. 11a), the standard deviation of the optical thickness σ τ 1 km computed from the 100 pixels of 100 m × 100 m (Fig. 11b), and the heterogeneity parameter ρ τ 1 km at 1 km × 1 km (Fig. 11c).We see that the difference BT 1 km is better correlated with σ τ 1 km (R=0.95).Indeed, as discussed previously, the PPA is the most important effect at this scale.We notice that the larger σ τ 1 km is, the greater BT 1 km is.Furthermore, as the increase in τ 1 km is not totally correlated with the increase in σ τ 1 km , the correlation between BT 1 km and τ 1 km is smaller, as is the correlation between BT 1 km and ρτ 1 km .The σ τ 1 km appears, therefore, to be the best parameter to highlight the heterogeneity effects and it will be used below to represent the field heterogeneity.
The BT 1 km as a function of the standard deviation σ τ 1 km for the different cirri of Table 1 and for the three channels of the IIR is presented in Fig. 12.The first remark on this figure is that the correlation coefficient R is better than 0.80 for all the cirri, except for cirrus 1.For this cirrus, the lowest optical thickness (0.45) associated with the strong scattering for different cirri presented in Table 1.R represents the average correlation coefficient over the three bands.at 8.65 µm lead to smoothing of the variability due to the photon horizontal transport.The five cirri have similar characteristics with different mean optical thickness τ c and heterogeneity parameter ρ τ .For each of them, BT3D 1 km and BT1 − D 1 km are computed 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 for the pixels presented in this study that are for optical thicknesses below 2. This figure shows that, for the same altitude, geometrical thickness and optical properties, BT 1 km depends mainly on the optical thickness sub-pixel 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 optical thickness heterogeneities for different ice crystal effective diameters D eff .In Fig. 13, we present BT 1 km as a 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 's decrease with the increase in D eff for the three TIR bands.For the bands at 10.60 µm (Fig. 13b) and at 12.05 µm (Fig. 13c), the single scattering albedo 0 and the asymmetry parameter g increase with D eff (Table 2); this leads to a decrease in the absorption inside each 1 km × 1 km pixel and an increase in energy in the forward peak.Therefore, photons emitted from the surface cross the cloud more easily, leading to a decrease in the contrast between large and weak optical thicknesses.The radiative field heterogeneities thus appear 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 's decrease.But between D eff = 20.09µm and D eff = 40.58µm, g increases slightly and 0 decreases.These two effects are opposite and the BT 1 km 's do not change significantly between these two effective sizes.
In summary, the amplitude of the optical thickness heterogeneity 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 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 obtain information on the influence of the crystal size on TOA BT.In reality, a large variety of particle sizes and shapes are present in cirri 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 realistic clouds, we employed the parametrization developed by Baran et al. (2009); Baran (2012); Baran et al. (2013) (see Sect. 2.2).The cirri 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).Note that the Field et al. (2007) parametrization is based on bulk measurements inside the cirrus and is not directly related to the cloud edges, where measurements can be different.However, as we showed, the microphysical properties have a slight influence on TOA brightness temperatures with regard to the IIR instrumental accuracy.Before studying the heterogeneity effects, 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. 14).The 1-D RT with homogeneous columns (BT1 − D 1 km ) is compared with the 1-D RT with heterogeneous columns (BT1 − Dhe 1 km ) for the CII-2 cirrus.We see that the difference is, on average, lower than 0.5 K for the three channels and it is a maximum for the band at 8.65 µm.Indeed, because the scattering is greater in this band and thus the absorption weaker, the vertical optical property variabilities have, therefore, a more significant impact on BT.
BT 1 km 's as a function of σ τ 1 km are presented in Fig. 15 in the three IIR channels for the CII-1 and CII-2 cirrus 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 over 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 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 heterogeneity effects on the TOA BT are, on average, lower than the IIR instrumental accuracy (1 K).The PPA and IPA bias have, thus, a limited impact in this case.Concerning cirrus CII-3 with homogeneous optical properties, the BT 1 km is slighter than for other cirri 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 absorbent in this band (Table 2).For the band at 8.65 µm, 0 is larger for small particles leading to more scattering which smoothes the radiative field heterogeneity.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) for IIR measurements.However, it could become significant for radiometers with a higher instrumental accuracy.

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 16 presents BT 1 km as a function of σ τ 1 km for cirrus 2 with the top altitude of 7.97 km and 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 leading to a stronger PPA bias.
To study the influence of the vertical extension, we then compare 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 16 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, forms less of a contrast with the surface temperature than cirrus 8 does, 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, on 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 (Tilted Independent Pixel Approximation (TIPA), Várnai and Davies, 1999) contrary to the 1-D framework which assumes infinite horizontal layers.
Figure 17 shows the BT 1 km averaged over all the pixels of the field ( BT 1 km ) as a function of the zenith view angle 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 v .For cirrus 1 to 5, BT 1 km is twice as large for v = 60 • than for v = 0 • .For CII-1, CII-2 and CII-3 cirrus, the difference is approximately ten times as large because they present a greater 3-D variability of the extinction coefficient.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 as a function of v .This is due to a saturation effect.These three cirri 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 bias tends to zero for these pixels as the surface emission is not visible any more.We also note that, for v = 0 • , BT 1 km is no longer correlated with σ τ 1 km , which is computed for integrated vertical cloud columns and, thus, does not represent the heterogeneity perceived from an oblique zenith view angle.Concerning view azimuth angle v , we see no real tendency, except for differences due to different crossed optical properties.

Influence of the observation scale on cloud heterogeneity effects
The previous results have been presented for the IIR spatial resolution of 1 km × 1 km.In this section, heterogeneity effects are presented for different spatial resolutions.Figure 18 shows the average brightness temperature difference BT as a function of the increase in the spatial resolution: 1 km, 2.5 km (only for cirrus 5 for time calculation reasons), 5 km, 10 km and 20 km (only for CII-1, CII-2 and CII-3 cirrus).For each spatial resolution, differences are computed between 3-D brightness temperatures, first calculated at 100 m and then averaged at the indicated spatial resolution, and brightness temperatures computed from the average of the optical properties at the scale of 1 km, 2.5 km, 5 km, 10 km or 20 km.The difference of brightness temperatures at each scale are then averaged over the whole cirrus to obtain BT .We observe that the BT is maximal for cirrus 5, the most heterogeneous, with a value about 1.5 K at 12.05 µm.We also notice that, BT quickly increases between 1 km, 2.5 km and 5 km and reaches an asymptote after that.This increase is all the more important as the cirrus mean optical thickness and mean heterogeneity parameters are large.

Summary and conclusions
This paper presents results concerning the impact of cirrus cloud heterogeneities on top-of-atmosphere brightness temperatures ( BT 1 km ).Spatial radiometers such as IIR/CALIPSO and MODIS/AQUA measure top-ofatmosphere brightness temperatures at 1 km × 1 km and their operational algorithms use the IPA and the 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 's depend mainly on the optical thickness standard deviation inside the observation pixel and on the brightness temperature contrast between the cloud top and the clearsky atmosphere.Furthermore, the impact of heterogeneity effects increases strongly with the zenith view angle except for cirri 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.
To summarize the different results presented in this paper, Fig. 19 shows the brightness temperature differences, BT 1 km , simulated at the nadir as a function of the optical thickness at 1 km × 1 km (τ 1 km ).The two red lines correspond to the instrumental accuracy of IIR.The different cases of cirrus are superimposed to obtain 2 000 pixels with different optical thicknesses.For these cirri, the contrast between the cloud top and clear-sky brightness temperatures is between -46 K to -67 K and we observe that the heterogeneity effects start to be significant ( BT 1 km larger than the IIR instrumental accuracy 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 cirri defined by Sassen and Cho (1992).In terms of σ τ 1 km , heterogeneity effects are greater than 1 K for σ τ 1 km ∼ 1.
We have also shown that heterogeneity effects increase strongly with the decrease of the spatial resolution until 10 km.Numerical weather prediction models assimilate cloudy radiances at 10 km resolution; it would be interesting to study the possibility of correcting the heterogeneity bias using, for example IIR, MODIS or Spinning Enhanced Visible and Infrared Imager (SEVIRI) information at 1 km.
The slight photon transport effect in the thermal infrared involves that heterogeneity effects are mainly due to the PPA bias.This bias should be easier to correct than 3-D effects as it could be estimated with subpixel heterogeneity or it could be decreased by reducing pixel size.However, photon transport effects increase when pixel size is reduced.For the IIR accuracy of 1 K, they become significant for spatial resolutions below 250 m.Assuming a 1 K radiometer accuracy, a spatial radiometer with a spatial resolution of 250 m × 250 m could retrieve cloud optical properties with weak PPA bias and weak 3-D effects.
In a future companion paper (Fauchez et al., 2014), we will focus on the estimation of the impact of cirrus heterogeneities on the cloud properties retrieved from IIR thermal infrared measurements with the 1-D approximation.

Figure 1 .Figure 2 .
Figure 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.

Figure 3 .
Figure 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) as a function of the frequency in Hz .

Figure 4 .
Figure 4. Observed ((a), (b)) and simulated ((c), (d)) cirrus cloud on 25 May, 2007 during the CIRCLE II campaign.(a) shows the MODIS "true color RGB" picture of the cirrus.The bold yellow line represents the CALIOP track and the red line the French Falcon 20 flight.Measurements start from point A to finish at point H.The black rectangle represents the cirrus area without any low-water cloud below.(b) represents the vertical profile of the attenuated backscattering coefficient of the cirrus observed by the lidar CALIOP/CALIPSO inside the black rectangle in (a).(c) represents the 20 km × 20 km optical thickness field generated at 12.05 µm with a horizontal spatial resolution of 100 m and with the mean optical thickness τ c = 0.41 observed by IIR at 12.05 µm.(d) represents the x-z view of the cirrus IWC generated by 3DCloud with a vertical resolution of 58 m.
Figure 5. (a), (b) and (c): vertical variation of the mean extinction coefficient σ e ; (d), (e) and (f): vertical variation of the single scattering albedo 0 ; and(g), (h) and (i): vertical variation of the asymmetry factor g for the three channels at 8.65 µm, 10.60 µm and 12.05 µm and for the cirrus CII-1 (blue curves) and CII-2 (red curves).

Figure 7 .
Figure 7. Variation of the 3-D brightness temperatures BT 3D 100 m as a function of the optical thickness at 100 m × 100 m τ 100 m for the three IIR channels and for cirrus 5. BT3D represents the average of BT3D 100 m , corresponding to two τ 100 m , BT3D represents the 3-D brightness temperature corresponding to the mean optical thickness τ .The mathematical formulation of the PPA due to the Jensen inequality is expressed by BT3D > BT3D( τ ).

---
): We plot the absolute value of the extinction vertical heterogeneity effect |BT1 − Dvhe 100 m − BT1 − D 100 m | in green, with BT1 − Dvhe 100 m corresponding to 1-D radiative transfer with independent column and vertically heterogeneous extinction coefficient.We plot the absolute value of the BT difference due to the horizontal transport which is computed with the calculation of |BT3Dvho 100 m − BT1 − D 100 m | in red, with BT3Dvho 100 m corresponding to 3-D radiative transfer with vertically homogeneous extinction.We plot the absolute value of the difference | BT 100 m | obtained with the calculation of |BT3D 100 m − BT1 − D 100 m | in blue.Note that | BT 100 m | is not the sum of the two effects described above because some of them can be opposing.

Figure 8 .
Figure 8. Absolute brightness temperature differences at 100 m × 100 m BT 100 m (in blue) due to the horizontal transport (in red), the extinction vertical variability (in green) and the statistical error of the code (black line) for cirrus 5. Results are presented for the bands at 8.65 µm, 10.60 µm and 12.05 µm.

Figure 9 .
Figure 9. Brightness temperatures at the scale of 100 m (BT 100 m ) as a function of the distance from the left cloud edge for a step cirrus cloud.The bold green line corresponds to the optical thickness at the scale of 100 m (τ 100 m ); straight lines correspond to 1-D brightness temperatures and curves to 3-D brightness temperatures.Bold lines refer to brightness temperatures of the geometrically thin cirrus ( Z = 0.4 km) and dashed lines to the geometrically thick cirrus ( Z = 2.0 km); blue, red and black colors correspond to 8.65 µm, 10.60 µm and 12.05 µm, respectively.

Figure 10 .
Figure10.Absolute brightness temperature differences at 1 km × 1 km BT 1 km (in blue) due to the horizontal transport (in red), the extinction vertical variability (in green), the statistical error of the code (black line) and the IIR instrumental accuracy (pink line) for cirrus 5. Results are presented for the bands at 8.65 µm, 10.60 µm and 12.05 µm.

Figure 12 .
Figure12.Brightness temperature differences, BT 1 km , viewed at the nadir as a function of the optical thickness standard deviation σ τ 1 km for different cirri presented in Table1.R represents the average correlation coefficient over the three bands.

Figure 13 .
Figure 13.Brightness temperature differences, BT 1 km , viewed at the nadir as a function of the optical thickness standard deviation σ τ 1 km for the same three cirrus fields with different ice crystal effective diameters: cirrus 3 (D eff = 9.95 µm), cirrus 6 (D eff = 20.09µm) and cirrus 7 (D eff = 40.58µm).

Figure 14 .
Figure 14.Differences between brightness temperatures viewed at the nadir between a 1-D radiative transfer with vertically heterogeneous columns (BT1 − Dhe 1 km ) and with homogeneous columns (BT 1 − D 1 km ) as a function of the optical thickness standard deviation σ τ 1 km for cirrus CII-2 and for the bands at 8.65 µm, 10.60 µm and 12.05 µm.

Figure 17 .
Figure 17.Mean difference BT 1 km as a function of the zenith view angle v ( • ) for cirrus 1 to 5 and cirrus CII-1 & CII-2 for the bands at 8.65 µm, 10.60 µm and 12.05 µm.

Figure 18 .Figure 19 .
Figure 18.Brightness temperature differences BT at 12.05 µm as a function of the spatial resolution for different cirri. .

Table 2 .
Optical parameters for aggregate ice crystals of the P. Yang model used in cirrus 1 to 8 and CII-3.