In the context of global climate change, the representation and role of clouds are still uncertain. Cirrus clouds cover between 15 and 40 % of the Earth's surface and play an important role in Earth's climate and radiative budget . The temperature difference between the cirrus cloud top and the Earth's surface leads to a warming of the atmosphere by cirrus clouds, capturing a part of the infrared radiation emitted by the atmosphere and surface. Also, cirrus clouds reflect part of the incident solar radiation into space, but this albedo effect is generally negligible for high thin clouds. Thus, on average, cirrus clouds lead to a positive radiative effect (e.g., a greenhouse effect), except for cirrus with large optical thicknesses (greater than 10; ) or at low altitudes (below 8 km in the tropics; ). The radiative impact and evolution of cirrus clouds depends on numerous factors such as cloud altitude, cloud optical and geometrical thickness, crystal shape, and effective size. Consequently, we need to improve our knowledge by improving the retrieval of cirrus cloud optical properties.

Global satellite observations are well suited to follow and better understand cloud evolution and characteristics. Therefore, many satellites are dedicated to their observations from microwave (wavelength of few millimeters) to visible ranges (wavelength up to 0.4 µm). Cirrus optical thickness (COT) and ice crystal effective diameter (CED) can be retrieved from radiometric measurements using dedicated operational algorithms. Many of these operational algorithms are developed for solar-reflectance channels, like that of the Moderate Resolution Imaging Spectroradiometer (MODIS), for the MOD06 product or the Clouds and the Earth's Radiant Energy System (CERES) product or the Visible Infrared Imager Radiometer Suite (VIIRS; ). Thermal infrared (TIR) channels are currently used in the MOD06 dataset to retrieve cloud temperature–pressure–altitude (and in other datasets to retrieve ozone concentration and clear-sky temperature–moisture information). However, several studies have shown that cirrus optical properties may be retrieved with a better accuracy using a combination of TIR channels instead of visible and near-infrared (VNIR) channels (such as the Nakajima and King method, ), as long as the cirrus is optically thin enough (with a visible optical thickness between roughly 0.5 and 3) and the CED is smaller than 40 µm. For example the split-window technique applied to the Advanced Very-High-Resolution Radiometer (AVHRR; ) and the Imaging Infrared Radiometer (IIR) onboard CALIPSO is used to retrieve CED and COT from the brightness temperature difference of two different channels in the infrared atmospheric windows where gaseous absorption is small. Based on the same spectral information, an optimal estimation method (OEM; ) is used for the Atmospheric Infrared Sounder V6 (AIRS, ) and in the research-level code of for MODIS. Another advantage of the TIR is that measurements can also be obtained in nighttime conditions, which gives a distinct benefit compared to solar-reflectance channels for developing ice cloud climatologies. However, in both VNIR and TIR optical property retrieval methods, each pixel is considered independent of its neighbors (independent pixel approximation, IPA; ) and fully homogeneous (plane-parallel homogeneous approximation bias, PPHB; ). These approximations are mostly due to time constraints on 3-D forward radiative calculations, the lack of knowledge about the sub-pixel variability and the 3-D structure of the cloud.

Many studies have been conducted in the solar spectral range to better understand the impact of cloud heterogeneities on cloud products. These studies primarily concern warm clouds such as stratocumulus and show that the sign and amplitude of retrieval errors depend on numerous factors, such as the spatial resolution, wavelength, geometry of observation and cloud morphology. In the TIR and for ice clouds, show that radiative transfer (RT) calculations using IPA can change the mean top of the atmosphere (TOA) radiative fluxes by 45 W m-2 in the shortwave and by 15 W m-2 in the longwave. show that the broadband thermal cooling rates are increased by around 10 % in 3-D RT by comparison to 1-D RT. Concerning infrared (IR) radiances or brightness temperatures (BTs), show that heterogeneity effects can significantly influence cirrus optical property retrievals at the 1 km scale of IIR thermal infrared observations, with potentially more than +10 K on TOA BT for heterogeneous pixels, depending also on the cloud altitude. also show that these TOA BT effects result in an overestimate of the retrieved effective diameter by more than 50 % for small crystals (CED under 20 µm) and underestimate the retrieved optical thickness by up to 25 %. These errors could significantly influence the cirrus feedbacks assumed in global atmospheric models.

The impact of cloud horizontal heterogeneity on both TOA radiation and retrieved products depends on the spatial resolution of the instrument (pixel size) and the cloud type. For example, show for stratocumulus clouds that the heterogeneity impact on cloud optical thickness retrieved from nadir-visible radiometric measurements is at a minimum around a pixel size of 1 or 2 km for a small solar zenith angle (22.5∘). Higher spatial resolutions enhance the IPA error (IPAE), which increases when the photon mean path (before absorption or cloud escape) is equal to or larger than the spatial resolution. Here we refer to the word photon in the sense of a fictive light particle (FLIP; ) for stochastic Monte Carlo simulations. Lower spatial resolutions have larger errors due to the homogeneous and plane-parallel cloud assumption bias, which increases when the assumed-homogeneous pixel size is increased. These results are very relevant as they can allow us to estimate the average error due to cloud heterogeneity based on the spatial resolution of any spaceborne radiometer. In addition, studies such as that attempt to identify the spatial resolution at which the error is at a minimum can help to define the ideal spatial resolution for future instruments. However, because such studies focus only on stratocumulus clouds, which are very different from cirrus, and because they were only conducted for visible wavelength, their conclusions cannot simply be extrapolated. In this paper, we focus our attention on cirrus clouds by simulating MODIS nadir TIR measurements (at the 8.52, 11.01, 12.03 and 13.36 µm wavelengths) and compare the impact of horizontal cloud heterogeneity as a function of spatial resolution from 50 m to 10 km. In Sect. , we present the 3DCLOUD model used to simulate a realistic cirrus case study and then we discuss the ice crystal optical property model used in MOD06, as well as the 3DMCPOL radiative transfer code used to simulate the 3-D RT inside the atmosphere in the thermal infrared. In Sect.  we describe the PPHB and IPAE that impact the TOA BT at nadir for our cirrus cloud. In Sect.  we study the impacts of cirrus heterogeneities on TOA BTs viewed from nadir as a function of the spatial resolution for the four MODIS TIR channels above. The influence of the geometry of observations is discussed in Sect. . Summary and conclusions are given in Sect. .

Clouds have variabilities at many different scales. However, in retrieval algorithms, for simplicity and computational reasons, the independent column approximation (ICA; ) is commonly applied; cloud layers are assumed to be vertically and horizontally homogeneous with an infinite horizontal extent (i.e., independent of each other). From the satellite retrieval point of view, the ICA is often named IPA for independent pixel approximation . Obviously, in reality, the pixel is not homogeneous and the radiative transfer between cloudy columns occurs in 3-D. Comparisons of BT simulated with these two RT approaches (IPA and 3-D) allow us to highlight the cloud heterogeneity effects on BT.

We simulated BT with 3DMCPOL at scales ranging from 50 m to 10 km. For each scale, BT values are computed using the 1-D RT assumption at the averaged COT (BTkm1-D with x as the scale and kilometers as the distance unit) and compared with 3-D radiance simulations at the finest field spatial resolution (50 m), arithmetically averaged to the scale being considered and converted to BT (for simplification reasons, we will refer to this process as BT averaging). The latter are noted as BT50m3-D‾xkm. The choice of the native spatial resolution for 3-D computations should be much smaller than the photon mean path (distance traveled before absorption or cloud escape) to account for horizontal radiative transport effects. However, the finer the spatial resolution is, the more pixels can communicate. The finer spatial resolution 3DMCPOL can achieve in a reasonable computational time for 10 km domain is 50 m. Table  summarizes the number of scattering and photon mean paths computed using (chapter 12) for various optical thicknesses and for channels centered at 8.52, 11.01, 12.03 and 13.63 µm. Note that the number of scatterings increases with the optical thickness and is almost twice as large at 8.52 µm. Obviously the photon mean geometrical path decreases with optical thickness (for the same cloud geometry) and is about 3 km at 8.52 µm for an optical thickness of 1 and only about 0.5 km for an optical thickness of 10.

In Fig. , we plot 1-D and 3-D BTs as a function of the optical thickness at 12.03 µm and at the spatial resolution of 50 m (panel a), 250 m (panel b), 1 km (panel c) and 5 km (panel d) for the four MODIS TIR channels. For a better readability, 1-D cases are a color tone lighter than their corresponding 3-D case. First, we see that 3-D and 1-D BTs decrease as optical thickness increases because the warmer surface contributes less to the signal as the cloud becomes more opaque. Also, the relationship between the BT and the optical thickness is nonlinear and depends on the optical thickness. This is particularly clear for the highest spatial resolution (50 m) where no averaging of the 3-D BTs has been performed. Two effects explain the differences between 3-D and 1-D BTs: the plane-parallel and homogeneous bias and the horizontal radiative transport.

As discussed in Sect. , heterogeneity effects on the radiative fields observed from nadir at TOA depend, on the one hand, on the sub-pixel optical thickness inhomogeneity (PPHB) and on the other hand, on the IPAE (HRT effect). The optimal resolution for cloud retrievals is therefore a compromise between reducing the PPHB by improving the spatial resolution but not causing larger increases in HRT effect. The objective is thus to find the smallest spatial resolution that strikes a balance between the PPHB and the absolute error due to the HRT. This spatial resolution depends of course on the wavelength (dependence on the photon mean path), cloud type (different optical properties, optical and geometrical thicknesses, and altitude) and the geometry of observation.

The total difference, computed as the total arithmetic mean difference (AMD) between averaged 3-D and non-averaged 1-D TOA BT viewed from nadir as a function of the spatial resolution, is given by AMDΔBTxkm3-D-1-D‾=∑i=1NBT50m3-D‾xkm-BTxkm1-D/N, with N the number of pixels at the spatial resolution x km. Three-dimensional BTs at all scales are estimated by averaging the 50 m BT to the x km scale, while 1-D BTs are directly computed at the x km scale at the averaged optical thickness. Because averaging BTs from a fine to a coarser spatial resolution will give a different result than BTs of the averaged optical thickness, we thus compared here how the nonlinearity between BT and optical thickness, as well as 3-D radiative effects, impacts TOA BTs at a given spatial resolution.

In order to separate the contribution of the PPHB and HRT to the total AMD, we also average the 50 m 1-D radiances to each x km scale. The PPHB is then the arithmetic mean difference between averaged 1-D and non-averaged 1-D TOA BT viewed from nadir as a function of the spatial resolution, and is given by PPHBΔBTxkm1-D-1-D‾=∑i=1NBT50m1-D‾xkm-BTxkm1-D/N.

Note that, because the PPHB is always positive or nil, ΔBTxkm1-D-1-D‾ is also either positive or nil. It is straightforward to be positive for the whole field, but locally, on the scale of a pixel, it can be either positive or negative, contributing to increase or reduce the AMD.

To highlight the absolute effect of the HRT, which can be considered as the mean deviation of the BT due to HRT, we also calculate the total mean absolute difference (MAD) between averaged 3-D and non-averaged 1-D TOA BT viewed from nadir as a function of the spatial resolution using the following equation: MADΔBTxkm3-D-1-D‾=∑i=1NBT50m3-D‾xkm-BTxkm1-D/N.

This is almost the same as Eq. () but for the sum of the absolute value of the difference. The mean deviation due to HRT at each spatial resolution is then obtained by subtracting the PPHB from the total MAD (|HRT| = MAD - PPHB). The MAD allows us to represent, at each spatial resolution, the mean deviation of the BT due to the cumulative effects of PPHB and |HRT|, and it is this parameter that we seek to minimize in order to estimate the optimal pixel size for IR cirrus retrievals.

Figure  shows the AMD and MAD in panel (a) and in panel (b) the PPHB and |HRT| of ΔBT‾ estimated at TOA from nadir for the whole cirrus field as a function of the spatial resolution for the MODIS TIR channels centered at 8.52, 11.01, 12.03 and 13.36 µm. In Fig. , we note that, for all channels, AMD is always smaller than MAD because the PPHB can be partially offset by the HRT when it is negative.

As the behavior is different for different channels, we discuss first the more absorbing channels (centered at 11.01, 12.03 and 13.36 µm) and then the more scattering channel (centered at 8.52 µm).

In the previous sections, results were shown for simulated observations from nadir. In this section, we discuss off-nadir observation geometries. In addition to the PPHB and HRT effects, another bias appears when looking off-nadir. Indeed, the oblique line of sight can cross many different cloudy columns in 3-D radiative transfer mode, while in 1-D, the cloudy column underneath a given pixel is considered horizontally infinite and thus fully containing the line of sight. We name this last bias the tilted homogeneous extinction assumption bias (THEAB). Note that the results of this section are strongly dependent on the cloud structure (with fallstreaks or not) and may be generalized to cirrus with similar patterns.

First of all, we can see in Fig.  that for off-nadir views, the PPHB is enhanced due to the increase in the curvature (non linearity) between BT and optical thickness with the view zenith angle. Note that we can also see in this figure that the saturation in BT with respect to changes in optical thickness appears earlier at Θv=60∘ (τ50m1-D∼7) than at Θv=30∘ (τ50m1-D∼8) and 0∘ (τ50m1-D∼9).

BTs differences between the viewing geometries can be seen in Fig.  with 1-D BT (left column) and 3-D BT (right column). In 1-D we can clearly see that increasing the viewing zenith angle reduces the average BT and that small differences appear for different azimuth view angles. In 3-D, because the line of sight can cross many different cloudy columns, the radiative field is much more dependent on both the zenith and azimuth view angles. The differences between the 1-D and 3-D fields for oblique views is mostly due to the THEAB.

Like the HRT effect, the THEAB is due to the IPAE and both effects are thus merged and represented in Fig.  by the IPAE. In Fig.  we can see the AMD (bold lines with squares), the MAD (bold lines with triangles), the PPHB (dashed line with crosses) and the IPAE (dotted line with stars) of (ΔBT‾) in panel (a) for viewing zenith angles Θv=0∘, 30∘ and 60∘ at a viewing azimuth angle of Φv=0∘, and panels (b) and (c) for viewing azimuth angles of Φv=0∘, 45∘, 90∘ and 180∘ at Θv=30∘ and 60∘ as a function of the spatial resolution for the channel centered at 11.01 µm only. In Fig. a, MAD(ΔBT‾) at Θv=30∘ and 60∘ for spatial resolutions between 1 and 10 km are larger than at nadir because of the larger PPHB as seen in Fig. .

In Fig. a, the mean absolute difference MAD(ΔBT‾) at Θv=30∘ and 60∘ is very large below ∼1 km due to the fact that the line of sight crosses many different columns in 3-D (large THEAB, which contributes strongly to the IPAE represented by the dashed lines with stars). However, when looking at the arithmetic mean difference AMD(ΔBT‾), we see that at Θv=60∘, it is negative for spatial resolutions below 500 m due to two effects: (i) first, in 3-D, due to the very oblique view, the line of sight crosses many cloudy columns of various optical properties for which the extinction is summed and leads, in most cases, to large optical paths. Such large optical paths imply that the top of the cirrus mostly contributes to the TOA BT. In contrast, some lines of sight cross small optical thicknesses at which photons emitted from the surface, warmer than the cloud, contribute to the TOA BT. This leads to 3-D BT being smaller than 1-D BT on average, and thus to negative AMD(ΔBT‾) values. We can also see this in Fig. f where the blue color (cold emission temperature at the cloud top) is more present than in 1-D (Fig. e). (ii) In 3-D, the line of sight crosses so many different columns that the difference between nearby lines of sight is reduced and the heterogeneity of the BT field is smaller in 3-D (SD[BT]∼20.2 K) than in 1-D (SD[BT]∼22.3 K) as we can see in Fig. f by comparison to Fig. e, respectively. Then, when 50 m BT values are averaged following Eq. (), 1-D BTs are increased more by the PPHB than 3-D BTs are, which contributes to the overall tendency of 1-D BT > 3-D BT and to the negative value of AMD(ΔBT‾).

While both effects are particularly strong below 500 m where the pixel size is small, they occur at every spatial resolution, explaining why PPHB is always larger than the AMD due to the IPAE. Note that, contrary to the PPHB, the THEAB contribution to the IPAE does not increase monotonously with Θv because it is related to the heterogeneity of the extinction along the line of sight for 3-D computations at 50 m, which can be smaller at Θv=60∘ than at Θv=30∘. We can also see that the IPAE is negative and the AMD is positive at 2.5, 5 and 10 km at Θv=60∘. Knowing that the HRT effect does not impact BT because the pixel size is too large, the IPAE is essentially due to the THEAB. In contrast to the higher spatial resolutions, the number of cloudy columns crossed by the line of sight is small and the large averaging homogenizes the field and thus reduces the AMD to a level close to or equal to the MAD. As a result, the AMD is smaller than the PPHB, which means that at coarse spatial resolutions the PPHB clearly dominates and the AMD is reduced, and not amplified, by the IPAE. Since AMD = PPHB + IPAE, the IPAE is negative as PPHB is larger than AMD or MAD. For Θv=60∘, the conclusions are similar to those for Θv=30∘, but with larger differences due to the greater IPAE between 3-D and 1-D BTs.

Concerning the change of the viewing azimuth angle at Θv=30∘ and 60∘, the difference in AMD, MA , PPHB and IPAE between the four angles is quite small except at Φv=45∘. Indeed, at this azimuth angle, the line of sight is parallel to the cirrus fallstreaks as we can see in Fig. g and h for Θv=60∘. Therefore, the variability along the oblique line of sight is weaker, reducing the smoothing effect of the 3-D field, which is closer to the 1-D field averaged heterogeneity (SD[BT]∼21.1 K in 3-D and SD[BT]∼22.3 K). In addition, the line of sight can pass only optically small paths and result in large BTs just as in 1-D. As a result, MAD(ΔBT‾) at Φv=45∘ is reduced at spatial resolutions at which fallstreaks are still observable (≤ 2.5 km). Above this value, the spatial resolution is so low that the fallstreaks are smoothed and the effect disappears.

Off-nadir, it is not easy to determine the spatial resolution at which the absolute value of ΔBT‾ occurs due to the combined heterogeneity and 3-D effects reaching a minimum because the spatial resolution depends on the viewing angle as well as on horizontal and vertical inhomogeneity. However, looking at Fig.  we can say that this location is at a coarser resolution than at nadir, as the THEAB drastically increases the ΔBT‾, especially at high spatial resolutions. The spatial resolution at nadir where the AMD of ΔBT‾ is the most mitigated for nadir view therefore sets a lower limit for off-nadir viewing geometries on both the AMD of ΔBT‾ and the spatial resolution at which the combined effects are minimum. These results were limited to the channel centered at 11.01 µm because computations for other channels were too computationally expensive. However, optical properties for channels at 11.01, 12.03 and 13.36 µm are close, leading to similar MAD(ΔBT‾) for nadir view as seen in Fig. 8. MAD(ΔBT‾) for other view angles should therefore be equivalent to the one at 11.01 µm. Only the 8.52 µm channel may have a different behavior. However, considering MAD(ΔBT‾) differences between 11.01 and 8.52 µm in Fig. 8, we can expect that MAD(ΔBT‾) for 8.52 µm will be larger for a smaller pixel size due to the larger scattering and the greater horizontal radiative transport.

The accurate remote sensing of cirrus clouds is very important in order to improve the parametrization of clouds in climate models and to better understand their role in the Earth–atmosphere system. Cloud heterogeneities may have a significant impact on the accuracy of retrieved cloud optical properties. In this work, we model the impact of cirrus cloud heterogeneities on top-of-the-atmosphere brightness temperatures as a function of the spatial resolution from 50 m to 10 km and at four MODIS thermal infrared channels centered at 8.52, 11.01, 12.03 and 13.36 µm. A three-dimensional cirrus cloud structure is modeled with the 3DCLOUD cloud generator and radiative transfer simulations are performed with the 3DMCPOL code. In this study, we assume that TOA brightness temperature differences between BT computation assuming 1-D RT inside a homogeneous pixel and 3-D RT inside a heterogeneous pixel depend on two effects: (i) the optical thickness horizontal inhomogeneity leading to the plane-parallel approximation bias and the (ii) horizontal radiative transport effect due to the independent pixel approximation error. The cloud vertical heterogeneities of optical properties are neglected here, based on the findings of . As previous studies already showed, the PPHB is the larger heterogeneity effect for nadir observations at the typical spatial resolution of polar orbiters such as AIRS, MODIS or IIR. The PPHB impacts mainly the low spatial resolutions, while the IPAE impacts mainly the high spatial resolutions. Although, due to the IPAE, the amplitude of the error in BT can be large at high spatial resolutions, the difference between the errors for different channels is quite small in comparison to the difference at coarse resolution. A similar error between channels can then mitigate the impact on the optical property retrieval. For our simulated cirrus case, we find that the approximate spatial resolution at which the PPHB and HRT effects lead to a minimum total effect at nadir is between 100 and 250 m. In order to extrapolate this result to different cirrus clouds, a sensitivity study has been conducted. The results show that changing the average cloud optical thickness affects the magnitude of the effects but does not significantly change the spatial resolution of the minimum. A spaceborne radiometer with a nadir spatial resolution between 100 and 250 m will allow the retrieval of cirrus optical properties in the thermal infrared with mitigated overall heterogeneity and radiative effects. In future studies, we will investigate how the errors on COT and CED retrievals due to horizontal inhomogeneities and 3-D effects are scale dependent.

Concerning off-nadir views, when Θv>0∘, the line of sight may cross several different cloudy columns in 3-D RT but not in 1-D RT, leading to the tilted homogeneous extinction assumption bias. This strongly increases the mean deviation between 3-D and 1-D BT, especially at fine spatial resolutions. However, on average, an increase in viewing zenith angle decreases the 3-D BT values as well as their heterogeneity, reducing the total error due to PPHB and IPAE. The dependence of the total effect on the azimuth angle could also be important for particular viewing orientations with respect to the cloud. For instance, the cloud heterogeneity, and thus the total effect, is smaller when the line of sight is parallel to the fallstreaks of the cloud, and is larger elsewhere. It thus seems that, for arithmetic field average values, the minimum total effect arises at nadir. Also, the THEAB leads to a shift in the spatial resolution of the minimum total effect toward coarser spatial resolutions. Off-nadir, it is clear that the horizontal and vertical structure of the cloud may change the conclusions. However, we have chosen the uncinus cirrus structure (with fallstreaks corresponding to intervals of thick and thin optical thicknesses), which is one of the most common among the variety of cirrus structures. We can thus extrapolate that results may be comparable to other uncinus cirrus, but may be different from others structures such as the patchy structures of cirrus floccus.

Note that these simulations were performed for a unique CED of 20 µm, very common in cirrus clouds but relatively small. However, for example, increasing CED to 80 µm leads to a convergence of the single-scattering albedo across all the TIR channels towards values between 0.5 and 0.6 (0.5 being the geometric optics limit). This implies less scattering and thereby horizontal transport in the 8.52 µm channel (ϖ0∼0.75 for CED =20 µm in this study). The differences between channels should thus be weaker and consequently the impacts on cloud optical property retrievals, which depend on the radiance relative difference between channels. Also, because single-scattering albedo values for all the channels at Deff=80 µm are close to those at 13.36 µm for Deff=20 µm used in this study, all the channels for Deff=80 µm will have a similar heterogeneity effect on TOA BT across spatial resolutions than for the 13.36 µm channel presented in this study. In Part 2 of this work we will study the impact of cirrus heterogeneities on visible and near-infrared MODIS channels and will make comparisons with the results of Part 1 here. We anticipate that the results will be different since 3-D effects are stronger for visible and near-infrared wavelengths and that solar geometries will play an important role. Additional perspectives will concern the impact of cirrus cloud heterogeneities on the optical property retrievals. Indeed, the dependence of heterogeneity and 3-D effects on the wavelength can be an issue for retrieval techniques using a combination of many wavelength ranges (such as optimal estimation methods). Other clouds, such as cumulus or stratocumulus, should also be considered because results are expected to be strongly dependent on the cloud type.