Corresponding author: Z. Zhang, Physics Department, University of Maryland, Baltimore County, Baltimore, MD 21250, USA. (firstname.lastname@example.org)
 This study investigates effects of drizzle and cloud horizontal inhomogeneity on cloud effective radius (re) retrievals from the Moderate Resolution Imaging Spectroradiometer (MODIS). In order to identify the relative importance of various factors, we developed a MODIS cloud property retrieval simulator based on the combination of large-eddy simulations (LES) and radiative transfer computations. The case studies based on synthetic LES cloud fields indicate that at high spatial resolution (∼100 m) 3-D radiative transfer effects, such as illumination and shadowing, can induce significant differences between retrievals ofre based on reflectance at 2.1 μm (re,2.1) and 3.7 μm (re,3.7). It is also found that 3-D effects tend to have stronger impact onre,2.1 than re,3.7, leading to positive difference between the two (Δre,3.7−2.1) from illumination and negative Δre,3.7−2.1from shadowing. The cancellation of opposing 3-D effects leads to overall reasonable agreement betweenre,2.1 and re,3.7 at high spatial resolution as far as domain averages are concerned. At resolutions similar to MODIS, however, re,2.1 is systematically larger than re,3.7when averaged over the LES domain, with the difference exhibiting a threshold-like dependence on bothre,2.1and an index of the sub-pixel variability in reflectance (Hσ), consistent with MODIS observations. In the LES cases studied, drizzle does not strongly impact reretrievals at either wavelength. It is also found that opposing 3-D radiative transfer effects partly cancel each other when cloud reflectance is aggregated from high spatial resolution to MODIS resolution, resulting in a weaker net impact of 3-D radiative effects onre retrievals. The large difference at MODIS resolution between re,3.7 and re,2.1 for highly inhomogeneous pixels with Hσ> 0.4 can be largely attributed to what we refer to as the “plane-parallelrebias,” which is attributable to the impact of sub-pixel level horizontal variability of cloud optical thickness onre retrievals and is greater for re,2.1 than re,3.7. These results suggest that there are substantial uncertainties attributable to 3-D radiative effects and plane-parallelre bias in the MODIS re,2.1retrievals for pixels with strong sub-pixel scale variability, and theHσ index can be used to identify these uncertainties.
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 Low-level maritime water clouds have a strong cloud radiative forcing [Hartmann et al., 1992] and are thought to be particularly sensitive to aerosol influences owing to their low altitude. A key parameter in aerosol-cloud interactions is the cloud droplet effective radiusre, which also has a strong influence on cloud radiative effects. In satellite-based retrievals,re is most frequently derived together with cloud optical thickness (τ) from reflectance measurements at two wavelengths [Nakajima and King, 1990], one usually in the visible or near-infrared spectral region (for example, 0.86μm), where water has negligible absorption and therefore cloud reflection is determined mainly by τ, and the other in the shortwave infrared (for example, 1.6 μm, 2.1 μm, or 3.7 μm), where liquid- and ice-phase water have significant absorption and cloud reflectance decreases with increasingre. This so-called bi-spectral method has been widely adopted [Han et al., 1994; Platnick and Twomey, 1994; Nakajima and Nakajima, 1995; Platnick et al., 2003; Roebeling et al., 2006] and forms the basis of global surveys of particle size by the MODIS instruments [Platnick et al., 2003]. Note that in both the operational MODIS cloud product and in this study, τ is defined with respect to the visible spectral region and remains almost invariant from the visible to shortwave infrared regions of interest in this study.
 Because of their strong radiative forcing [Hartmann et al., 1992], even small biases in re or τretrievals for low-level maritime water clouds can lead to significant errors in calculations of cloud radiative forcing. It is estimated that even a moderate perturbation to there of these clouds can lead to a global radiative forcing perturbation of around 1∼2 W m−2 [Oreopoulos and Platnick, 2008]. It is therefore critical to identify the sources and magnitudes of bias in the cloud property retrievals for these clouds.
 Recently, several studies [Nakajima et al., 2010b; Seethala and Horváth, 2010; Zhang and Platnick, 2011] have shown that re retrievals based on the 3.7 μm MODIS band (re,3.7) are systematically smaller than those based on measurements in the 2.1 μm band (re,2.1). As shown by Zhang and Platnick , the difference (Δre,3.7−2.1) is strongly dependent on cloud regime: over the trade wind cumulus cloud region, where clouds are often broken, the difference can be as large as 10 μm. These findings indicate substantial uncertainties in the current satellite-based cloudre retrievals.
 Two hypotheses have been proposed to explain these large differences. One possibility is the presence of drizzle within the cloud. The weighting function for cloud reflectance is a function of the magnitude of absorption [e.g., Platnick, 2000; Zhang et al., 2010], such that re,3.7 is weighted more toward cloud top than re,2.1 because of the stronger water absorption and smaller penetration depth of the 3.7 μm band. Drizzle in marine boundary layer clouds causes re to increase toward cloud base, a vertical structure that might potentially produce larger values of re,2.1 than re,3.7 [e.g., Chang and Li, 2002; Nakajima et al., 2010a, 2010b; Suzuki et al., 2010].
 The second hypothesis centers on the role of horizontal inhomogeneity, and possibly three-dimensional radiative transfer effects, in determining cloud reflectance [Boeke, 2009; Hayes et al., 2010; Seethala and Horváth, 2010; Zhang and Platnick, 2011]. Most satellite retrievals, including the bi-spectral method, assume that clouds are both horizontally and vertically homogeneous within the instrument field of view (typically on the order of a few hundred meters to a few kilometers). Unfortunately, this assumption can cause significant errors when the clouds exhibit sizable spatial variability within the sensor field of view [Marshak et al., 2006; Liang et al., 2009; Di Girolamo et al., 2010]. Zhang and Platnick  noticed that the re,3.7 and re,2.1 differ most in pixels with both large re,2.1and large sub-pixel inhomogeneity, and suggested that effects associated with cloud horizontal inhomogeneity may play an important role in causing the observed Δre,3.7−2.1 (see section 2 for details).
 It must be noted that the two hypotheses should not be considered mutually exclusive as they may both play a role in the observed Δre,3.7−2.1. Nevertheless, these hypotheses have very different implications. If Δre,3.7−2.1 is a result of drizzle, it could provide useful information, including remote drizzle detection [Chang and Li, 2002, 2003; Kokhanovsky and Rozanov, 2011]. If Δre,3.7−2.1 results from cloud horizontal inhomogeneity, however, it reflects the limitations of the retrieval algorithm assumptions and might be used to assess retrieval quality.
 Although an understanding of the relative importance of drizzle and cloud inhomogeneity on MODIS reretrievals is desired, it is difficult to separate the influences of these factors using MODIS observations alone. Fortunately, high-resolution cloud resolving models provide a useful tool for cloud remote-sensing studies. A high-resolution cloud resolving model, coupled with a plausible treatment of cloud microphysics, can provide detailed information on cloud macrophysical and microphysical structure. Cloud fields from such a model can be used as input to drive 3-D radiative transfer models to simulate satellite observations. Then, cloud property retrievals based on simulated observations can be compared with the original simulated cloud fields to identify the influence of various factors on passive cloud property retrievals. One of the advantages of such a simulator is its flexibility to accommodate various mechanisms at different levels of complexity. For example, the impact of 3-D effects can be estimated by comparing 3-D with 1-D radiative transfer simulations. The impact of drizzle can be estimated by artificially removing it from the simulated cloud fields. In previous studies, the difference betweenre,2.1 than re,3.7has been assessed and analyzed based on MODIS observations and hypotheses have been proposed to explain the difference. In this study, these hypotheses will be tested using synthetic cloud fields from a large-eddy simulation (LES) model. For this purpose, we have developed a MODIS cloud property retrieval simulator using a LES model combined with 1-D and 3-D radiative transfer models. Below we present several case studies based on this simulator and discuss the implications of the results.
 The objectives of this study are twofold. First, we will introduce our simulator and demonstrate its capabilities for simulating MODIS cloud property retrievals, including the subtle difference between re,2.1 and re,3.7. Second, using the simulator, we will investigate the effects of cloud horizontal inhomogeneity and drizzle on satellite reretrievals based on the bi-spectral method, their relative importance under various circumstances, and the implications for introducing systematic bias in the observational record. In particular, we will attempt to address the following questions:How and to what extent do drizzle and cloud inhomogeneity affect MODIS re,2.1retrievals? Do they affect the re,3.7retrieval in different ways or to different degrees? Finally, can they lead to systematic retrieval bias?
 We will first introduce in section 2the “plane-parallelre bias” using simple idealized cases, as well as summarize recent observational results with respect to differences between MODIS re,3.7 and re,2.1 for maritime water clouds. We will then describe the MODIS simulator in section 3 and the case studies in section 4. Results are discussed in section 5 in the context of MODIS observations.
 To provide clarity for more detailed discussions below, it is important to briefly introduce the concept of the plane-parallelrebias and revisit some 3-D radiative transfer effects known to have strong impacts on cloud property retrievals. A summary of the main findings from a recent study of MODISre retrievals for maritime water clouds follows.
2.1. Plane-Parallelre Bias
 We define the plane-parallelrebias as the impact of small-scale variability inτ on reretrievals that use area-averaged reflectance. This bias is illustrated using two idealized examples inFigure 1, which shows forward calculations of reflectance in a single non-absorbing band (0.86μm) and two bands with different amounts of liquid water absorption (2.1 μm and 3.7 μm). In Figures 1a and 1b, we assume that half of a MODIS pixel overlying a black surface is covered by a cloud with τ1 = 2.8 and re = 8 μm, and the other half is covered by a cloud with τ2 = 30.8 and re = 8 μm. Figures 1c and 1d assume the same optical thickness but use re = 18 μm. Focusing on the τretrieval, the figure illustrates the well-known “plane-parallel albedo bias” [Cahalan et al., 1994]: the retrieved τbased on the mean reflectance of inhomogeneous pixels tends to be smaller than the mean of the sub-pixelτ . In this example the value of τ retrieved from the averaged reflectance (10.8) is substantially smaller than the average value (16.8).
 This problem is more acute for retrievals of re because the reflectance used to infer re also depends on τover much of the range of plausible values. If the reflectance at non-absorbing and absorbing wavelengths depended only onτ and re,respectively, (in other words, the look-up-table is orthogonal) reflectance at absorbing wavelengths would be uniform in our example and particle size could be retrieved perfectly. As the figure demonstrates, however, the look-up-table is not orthogonal. The nonlinearity leads to a simultaneous underestimate ofτ(i.e., plane-parallel-albedo bias) and overestimate ofre(i.e., plane-parallelrebias). The area over which this is true is larger in the less-absorbing band, which explains why the size overestimate at 2.1μm is larger than at 3.7 μm (re estimates of 11.7 and 8.8 μm, respectively, in Figures 1a and 1b). The impact becomes more pronounced as re increases: in Figures 1c and 1d the true re = 18 μm while re,2.1 and re,3.7 retrieved from averaged reflectances are 24 μm and 18.1 μm, respectively, resulting in a Δre,3.7−2.1 around −6 μm.
 It is to be noted that the underlying assumption behind the examples in Figure 1 is that τhas stronger small-scale horizontal variability thanre. This assumption appears reasonable, as τ can vary over several order of magnitude, while re varies mostly from a few to a few tens of microns. On the other hand, if rehas substantial sub-pixel variability, the configuration ofτ and re would be different from those in Figure 1. The plausibility of the example shown in Figure 1 will be examined in the LES cases.
2.2. 3-D Radiative Transfer Effects
 Cloud horizontal inhomogeneity can also induce net horizontal transfer of radiation that is neglected by the one-dimensional models on which retrievals are based. Effects of this 3-D radiative transfer, such as illuminating and shadowing, are known to have potential impacts onre and τ retrievals [e.g., Várnai and Davies, 1999; Várnai and Marshak, 2002; Kato et al., 2006; Marshak et al., 2006]. For example, the illuminating effect makes clouds appear brighter than expected under the plane-parallel cloud assumption, resulting in overestimatedτ and underestimated re retrievals. The shadowing effect makes clouds appear darker, resulting in underestimated τ and overestimated re [Marshak et al., 2006].
 The 3-D radiative effects may affectre,2.1 and re,3.7retrievals to different extents and lead to significant differences between the two. In a simple step-cloud case,Zhang and Platnick noted that the 3-D radiative effects tend to have a stronger impact onre,2.1 than re,3.7 (see their Figure 15). This is likely because stronger absorption in 3.7 μm leads to less multiple-scattering and, as a result, less photon horizontal transport. The implication is that the illuminating effect tends to result in positive Δre,3.7−2.1 because it reduces re,2.1 more than re,3.7, while the shadowing effect tends to result in negative Δre,3.7−2.1 because it increases re,2.1 more than re,3.7.
 When a cloud pixel has strong horizontal heterogeneity, both 3-D radiative effects and the plane-parallelre bias discussed in the previous section may collude to create either positive or to either positive or negative Δre,3.7−2.1. However, because the illuminating and shadowing effects naturally come in pairs and tend to cancel each other [Marshak et al., 2006], the plane-parallelre bias might be expected to be the dominant factor when averaged over many cloud cells. We will return to this point in section 5.2.
where stdev[Ri(0.86 μm, 250 m)] and mean[Ri(0.86 μm, 250 m)] indicate the standard deviation and mean of the measured reflectances, respectively, for the principle sixteen 250 m-resolution sub-pixels within the 1 km MODIS retrieval footprint. Thus,Hσhas a spatial resolution (i.e., 1 km) consistent with the cloud property retrieval and increases with pixel inhomogeneity. The dependence of pixel-level Δre,3.7−2.1 on re,2.1 and Hσ in one month of MODIS observations of warm liquid phase clouds (with cloud top temperature >273 K) over ocean is illustrated in Figure 2. The most striking feature is the area of large differences (Δre,3.7−2.1 ∼ −10 μm) in the upper right corner of Figure 2. The large differences are associated with large (>25 μm) values of re,2.1and large sub-pixel inhomogeneity (Hσ ∼ 1). Δre,3.7−2.1 remains stable when Hσ is smaller than about 0.4, but becomes increasingly negative with increasing Hσ when Hσ > 0.4. For a given Hσ, Δre,3.7−2.1 generally increases with re,2.1.
 As mentioned earlier, some studies interpret negative values of Δre,3.7−2.1 as evidence of drizzle in water clouds, such that the 2.1 μm band (with weak water absorption) penetrates deeper into the cloud than the 3.7 μm band and so can be more strongly affected by large drizzle drops in the lower part of the cloud [Chang and Li, 2002; Nakajima et al., 2010a]. Others argue that the difference between re,2.1 and re,3.7is mainly a result of the plane-parallelre bias illustrated in Figure 1. Both hypotheses would find support in Figure 2. The fact that the strongest Δre,3.7−2.1 is found where re,2.1 is larger than 25 μm is consistent with the drizzle hypothesis because drizzling clouds tend to have larger rethan non-drizzling clouds. On the other hand, large values of the sub-pixel inhomogeneity indexHσin the same region would imply plane-parallelre bias in the retrieval.
3. Simulating MODIS Retrievals
 Since the two hypotheses cannot be distinguished using the observations alone, we turn to retrievals from synthetic clouds in which the radiative transfer effects can be carefully controlled. Cloud structure is obtained from simulations by the DHARMA LES model [Stevens et al., 2002; Ackerman et al., 2004]. Here, DHARMA uses a single-moment bin microphysics scheme to resolve size distributions of aerosols and activated water drops each into 25 size bins, spanning particle radii of 0.01–2.5 and 1–250μm, respectively. The size distributions of activated droplets on the DHARMA grid is used to derive cloud scattering properties at MODIS cloud bands, which in turn are used to drive radiative transfer simulations. The bin microphysics approach obviates the requirement of a priori assumptions about cloud drop size distributions. Fields from the LES model described by Feingold et al.  and Xue et al.  have also been examined, but are not shown here because the results are comparable to our analysis of the DHARMA cases.
 Radiative transfer is computed using the I3RC community Monte-Carlo model [Cahalan et al., 2005; Pincus and Evans, 2009] for 3-D simulations and the DISORT model [Stamnes et al., 1988] for 1-D simulations. The single-scattering properties of cloud droplets (extinction efficiencyQe,i, single-scattering albedoωi and phase function Pi) in each size bin, i, are computed using the Mie code of Wiscombe , following the steps described by Platnick and Valero , for each MODIS band used. In the DISORT model, the scattering properties of a cloudy grid cell are computed off-line by averagingQe,i, ωi and Piover the cloud drop size distribution (DSD) from the LES. In the I3RC model, the DSD-averaged scattering properties are constructed online (instead of pre-computed off-line as in the 1-D simulation), fromQe,i, ωi, and Pibased on the cloud DSD. The dynamical construction is implemented through a Monte-Carlo sampling of the relative cumulative extinction functionF defined as:
 For a given MODIS band, the shape of F is determined by the cloud DSD. In a scattering event, a random number ζ, uniformly distributed between 0 and 1, is generated. Cloud droplets in the ith size bin with Fi < ζ < Fi+1are then chosen to interact with (i.e., scatter or absorb) the photon. This on-online construction scheme preserves the detail of the cloud microphysical properties from the LES in radiative transfer simulation.
4. Case Studies
4.1. Cloud Fields From Large-Eddy Simulations
 Using the MODIS simulator described above, we have investigated the MODIS reretrievals for three large-eddy simulations of marine water clouds. The first (referred to as “ATEX clean” hereafter) and second (“ATEX polluted”) cases are based on an idealized case study [Stevens et al., 2001] from the Atlantic Trade Wind Experiment (ATEX), with different initial aerosol loadings. A diagnostic approach [Clark, 1974] is used for the aerosol, in which the total number concentration of unactivated aerosol particles plus activated droplets is fixed at values of 40 and 600 cm−3 for the clean and polluted cases, respectively, resulting in average cloud droplet concentrations (weighted by liquid water mixing ratio) of about 30 and 300 cm−3. The total number and size distribution of activated droplets vary in each grid cell over the course of the simulation. For the 8-h ATEX simulations the domain size is 9.6 km × 9.6 km × 3 km, with a uniform grid mesh of Δx = Δy = 100 m and Δz = 40 m. (Further details of the model setup for the ATEX cases are provided byFridlind and Ackerman .) A snapshot of the cloud field is taken every half hour over the last 4 h. Therefore, each case contains eight 3-D cloud scenes from the large-eddy simulation, equivalent to about 800 1 km × 1 km pixels. The third case (referred to as the DYCOMS-II case) is based on an idealization of conditions observed during the second research flight (RF02) of the Second Dynamics and Chemistry of Marine Stratocumulus project (DYCOMS-II). For this case, the domain size is 6.4 km × 6.4 km × 1.5 km, and the grid spacing is Δx = Δy = 50 m in the horizontal and stretched vertically, with a spacing minimum of 5 m near the surface and inversion. The average cloud droplet number concentration in the DYCOMS-II case is about 60 cm−3. A snapshot is taken every hour over the last 4 h of the 6-h simulation, yielding about 200 1 km × 1 km pixels. (Further details of the model setup for the DYCOMS-II case are provided byAckerman et al. .) Table 1 provides a summary of model setup and cloud properties for the three cases.
Table 1. Summary of Model Setup and Cloud Properties of the Three LES Cases Used for This Study
Figure 3 provides a planar view of the cloud τderived from LES for these three cases. The DYCOMS-II case is almost overcast, while the cloud fraction, defined as fraction of columns withτ > 0.1, in ATEX is around 0.7. Enough drizzle develops in the ATEX clean case that drizzle (i.e., r > 30 μm) accounts for a significant fraction of the total optical thickness τ, but is negligible in the other cloud fields. It is important to note that the cloud τ in Figure 3 is derived from the droplet number concentrations from the LES. It can be different from the τ retrieved from cloud reflectance. The three sets of cloud fields also differ significantly in terms of re and cloud inhomogeneity, which we will exploit to help understand the dependence of Δre,3.7−2.1 on cloud regimes.
Figure 4provides an example cloud microphysics simulation from the LES model. The cloud droplet size distributions at three different vertical levels of a selected LES column from the ATEX-clean case are plotted. The location of this column is indicated inFigure 3 (left) by a red star. This particular column is drizzling as indicated by the development of the drizzle mode in the size distribution from cloud top toward cloud base. Figure 4 also demonstrates the level of complexity of the LES cloud fields used in this study. In many previous studies based on LES cloud fields, cloud microphysical properties are often greatly simplified. For example, in Kato et al.  and Marshak et al.  a constant value of re is assumed for all cloudy cells. In contrast, in this study every detail of the bin microphysics from the LES model is preserved in the radiative transfer simulations as described in section 3.
4.2. Radiative Transfer Calculations
 Both 1-D and 3-D radiative transfer calculations are performed for the 0.86μm, 2.1 μm and 3.7 μm MODIS bands. Because most of the low-level maritime warm clouds of interest to this study are located in tropical or sub-tropical regions, reflectance is calculated at two solar zenith angles (SZA) (20° and 50°) and a single value of relative azimuth (30°) to simulate the high-sun and low-sun conditions frequently encountered by MODIS over the tropical or sub-tropical regions [Seethala and Horváth, 2010]. Reflectances under each SZA are computed at viewing zenith angles (10° intervals from −50° to 50°) that mimic the scan range of the MODIS instrument. Without explicit indication, retrieval results at different solar and viewing angles are simply combined together in the analysis for more robust statistics; the dependence of results on solar and viewing angles will be investigated later in section 5.2. For simplicity, the surface is assumed to be black and atmospheric absorption is not considered in any of the simulations.
 In practice the 3.7 μm band observation contains both solar reflection and thermal emission components, the latter having to be accounted for during retrieval. We initially included emission in our radiative transfer calculations; however, sensitivity studies (not shown) indicate that including the thermal correction makes no significant difference in the simulation, especially for pixels with τ > 5. For this reason, we will consider only the solar reflection component of the 3.7 μm observation.
 We perform retrievals using reflectance at high (native LES) and MODIS-like resolutions; for the latter we use area-averaged reflectance as input to the retrieval. (We have adopted 800 m resolution instead of the nominal 1 km because our model domains are more neatly divisible by 800 m. Sensitivity studies [not shown] indicate the difference between 800 m and 1 km retrievals is negligible.) Given two methods for computing radiative transfer (1-D and 3-D), this produces four sets of retrievals, each of which is affected by different factors (Table 2). High-resolution 1-D simulations, for example, are free of 3-D radiative effects and plane-parallelre bias, but are affected by the cloud vertical structure, including drizzle, so that differences between re,2.1 and re,3.7retrievals can be attributed to cloud physics. Retrievals simulated at 800 m resolution based on 3-D radiative transfer are affected by all potential factors and resemble real MODIS observations. It is worth pointing out that, no matter which method (i.e., 1-D or 3-D) or resolution (LES native or MODIS) are used in the forward radiative transfer simulations, to be consistent with the operational MODIS product the retrievals are based on the look-up-tables under plane-parallel cloud assumption. A simple threshold method based the cloud reflectance in the 0.86μm band (i.e., R(0.86 μm) > 2%) is used to mask the cloud pixels after radiative transfer simulation. Only pixels with fractional cloud cover exceeding 0.95 are used in the analysis.
Table 2. Summary of How Different Effects Influence the Retrievals Based on the Different Combinations of Spatial Resolution (LES Native or MODIS) and Radiative Transfer Schemes (1-D or 3-D)
3-D Radiative Effects
Cloud Vertical Structure and Drizzle Effects
1-D LES resolution
1-D MODIS resolution
3-D LES resolution
3-D MODIS resolution
 Before proceeding to retrieval results, it is important to point out that although a “reference re” is desirable for the comparison between re,3.7 and re,2.1, there are many ways (and therefore no unambiguous way) to define a reference re from the LES field. To appreciate this, one can consider the LES column in Figure 4. A reference re can be defined as re = 3LWP / (2τρw), where LWP is the cloud liquid water path of the column and ρw is the density of water. Such a reference re ensures the correct LWP when τ is known. However, this definition does not account for the fact that the MODIS 2.1 μm and 3.7 μm bands are only sensitive to the upper part of the cloud because of their limited penetration depth [Platnick, 2000]. One way to account for this sensitivity is to first derive a vertical weighting function that incorporates both the underlying cloud microphysical structure and the sensitivity of each MODIS band to that structure. Then a reference re consistent with the MODIS re retrieval mechanism can be derived, based on this weighting function, as described by Platnick . But such a reference reis equivalent to the band-specificreretrieved from 1-D radiative transfer simulations at LES resolution. Therefore, we simply use thereretrieval results from 1-D radiative transfer simulations at LES resolution as the referencere.
4.3. Results From the ATEX Clean Case
Figure 5 shows comparisons of re,2.1 and re,3.7retrievals for the ATEX clean case at the LES resolution based on 1-D (Figures 5a and 5b) and 3-D (Figures 5c and 5d) radiative transfer simulations. To be consistent with the observational results in Figure 2, only those pixels with cloud optical depth τ > 5 are included in Figure 5.
 There is relatively little variability in reference re in this simulation: most values lie between 13 μm and 22 μm, with very little difference between estimates from 2.1 μm reflectance and those from 3.7 μm reflectance (Figures 5a and 5b). The close agreement between re,3.7 and re,2.1 in Figure 5a suggests that drizzle or cloud vertical structure has either little impact, or affects reflectance at both wavelengths roughly equally. We return to this point in section 5.1.
 In 3-D simulations retrieved drop sizes agree well in the range between 15μm to 20 μm, but re,3.7 overestimates re,2.1 for re,2.1 < 15 μm and underestimates re,2.1 for re,2.1 > 20 μm. Most reference values of both re,3.7 and re,2.1 are between 15 μm and 20 μm, so these out-of-range retrievals (i.e.,re,2.1 < 15 and re,2.1 > 20 μm) are likely the result of 3-D effects. The results are also consistent with the idea that retrievals in the more-strongly absorbing 3.7μm band are less affected by 3-D effects than the 2.1μm band. Under this hypothesis, for example, when 3-D effects lead to an overestimate of truere, the overestimate in re,2.1 should be larger than in re,3.7, yielding negative Δre,3.7−2.1. This hypothesis will be further explored in section 5.2.
Figure 5indicates that, though 3-D radiative transfer increases the scatter, retrievals made from reflectance computed at high resolution are broadly consistent across wavelengths. Results at the MODIS resolution, shown inFigure 6, tell a different story: re,3.7retrievals in both 1-D and 3-D simulations are biased significantly smaller compared with there,2.1 retrievals. Note also that some very large (>20 μm) re,2.1 retrievals emerge in Figure 6a; Δre,3.7−2.1 can be up to −10 μm in these pixels. The 3-D retrieval results inFigures 6c and 6dare quite similar to the 1-D results, suggesting the same mechanism is operative in both 1-D and 3-D retrievals.
 Our results are also consistent with the MODIS observations shown in Figure 2. Figure 7 shows composite plots of MODIS resolution Δre,3.7−2.1 as a function of re,2.1 and Hσfor both 1-D and 3-D simulations of the ATEX clean case. In the 1-D retrieval (Figure 7a), re,2.1 is mostly between 15 μm and 20 μm and Δre,3.7−2.1 remains close to zero until Hσ reaches 0.5, after which re,2.1 rapidly jumps to much larger values and Δre,3.7−2.1 drops dramatically from near zero to negative values as large as −10 μm. Consequently, the largest negative values of Δre,3.7−2.1 are found at the upper right corner of Figure 7a, where both re,2.1 and Hσare large. The 3-D retrievals (Figure 7b) behave much like the 1-D results (Figure 7a), though with more spread and with a less defined threshold. Unique to the 3-D simulations is the transition of Δre,3.7−2.1 from red (positive) to green (negative) as re,2.1 increases from 12 μm to more than 20 μm. Positive values of Δre,3.7−2.1 correspond to points above the zero line in Figure 6d and above zero in Figure 5d. As discussed above, these points are likely attributable to 3-D radiative effects, such as photon horizontal transport, illuminating and shadowing.
4.4. Results From the ATEX Polluted and DYCOMS-II Cases
 Results from the ATEX polluted and the DYCOMS-II cases are consistent with these findings (Figure 8). By design, drop sizes are smaller in the ATEX polluted case (Figures 8a and 8b) than in the ATEX clean examples, with reference re values generally below 10 μm. Most pixels in the ATEX polluted case have small-to-moderateHσ < 0.5 and stable values of re,2.1 and Δre,3.7−2.1, with larger re,2.1 and large negative Δre,3.7−2.1 for the small population of pixels with Hσ > 0.5.
 The DYCOMS-II case (Figures 8c and 8d) is almost overcast and the clouds are more homogenous, so that Hσ is small everywhere in the domain and neither the threshold behavior of re,2.1 nor very large negative Δre,3.7−2.1 values (i.e., Δre,3.7−2.1 < −5 μm) are evident. The effects of 3-D radiative transfer are similar to those demonstrated for the ATEX clean case. In particular, 3-D effects have a greater impact onre,2.1 than on re,3.7 so that anomalously small values of re,2.1 (here, less than 14 μm) are associated with positive Δre,3.7−2.1, while pixels with abnormally large re,2.1 > 18 μm tend to have negative Δre,3.7−2.1.
Figures 7 and 8 demonstrate that differences between re,3.7 and re,2.1retrievals at MODIS resolution in our simulations are largely controlled by the sub-pixel level cloud inhomogeneity: Δre,3.7−2.1 becomes increasingly negative as Hσ increases, and the differences are associated with large Hσ ∼ 1, and large re,2.1 retrievals. These results agree well with the MODIS observations shown in Figure 2.
5. Disentangling the Impacts of Drizzle and Cloud Inhomogeneity
 In this section we investigate the effect of drizzle and sub-pixel level cloud inhomogeneity on MODISre retrievals, focusing on pixels with the largest magnitude of Δre,3.7−2.1 to investigate causal links among re2.1, Hσ and Δre,3.7−2.1. We use the ATEX clean case because of its significant drizzle and general consistency with the other two cases.
5.1. Influence of Drizzle
 We define drizzle drops as those droplets with r > 30 μm, as this threshold generally separates the droplet size distribution into drizzle and cloud modes (see Figure 4). We investigate the impact of these drops on our retrievals by performing calculations in which these drops are removed. Comparisons of LES resolution τ, re2.1, and re3.7retrievals based on 3-D reflectances between results that include or exclude drizzle drops are shown inFigures 9a, 9b and 9c, respectively. The MODIS resolution retrievals after the removal of drizzle are shown in Figures 9d and 9e in the same manner as in Figures 6d and 7b. The removal of large drizzle drops in the radiative transfer simulation results in slightly smaller cloud optical thickness (Figure 9a) and effective radius retrievals (Figures 9b and 9c), as might be expected, but the differences between simulations with and without drizzle are very small. Indeed, at the MODIS resolution (Figures 9d and 9e), the retrievals after the removal of drizzle look almost identical to those with drizzle (Figures 6d and 7b). Thus drizzle drops with r > 30 μm have a very minor impact on the re2.1 and τ retrievals in this particular example, so drizzle cannot be the primary reason for bias between re,2.1 and re,3.7. Similar findings are reported by Zinner et al.  and Painemal and Zuidema .
 We emphasize, however, that the above results are based on a particular LES case, which is not representative of all MODIS observations. For example, a significant fraction of MODIS pixels in Figure 2 have relatively small Hσ ≈ 0.1 and large Δre,3.7−2.1 ≈ −5 μm. However, none of our LES cases yields any retrieval in this region. The impact of drizzle for the pixels in this region therefore remains unclear and needs to be elucidated in future research.
5.2. Influences of Cloud Horizontal Inhomogeneity
 We now turn our attention to the plane-parallelrebias and 3-D radiative transfer effects. Using the ATEX-clean case we will show that both of these effects exist, and will demonstrate their influence there retrieval. We will then analyze their relative role in causing the Δre,3.7−2.1.
 The idealized cases in section 2.1illustrated how cloud horizontal inhomogeneity can result in a plane-parallelre bias and significant difference between re2.1 and re3.7. This bias is examined here in a more realistic setting using a pixel from the ATEX clean case. Figure 10 shows the cloud extinction coefficient and cloud reflectance along the cross section at SZA of 20° and viewing zenith angle of 0°. The vertical lines indicate the location of the MODIS resolution pixel analyzed in Figure 11. The 1-D and 3-D simulations of cloud reflectance are in general agreement (i.e., net horizontal transport is small) in all but a few regions inFigure 10b in which R(0.86 μm) based on 3-D simulation is substantially smaller than that based on 1-D simulation. The selected MODIS resolution pixel (i.e., 800 m) consists of 64 LES resolution sub-pixels (i.e., 100 m), the reflectances of which are mapped (blue asterisks) onto the retrieval LUTs inFigure 11. The values are clustered around the re = 19 μm contour line of the LUT. The red diamonds in the figure indicate the locations of the MODIS resolution cloud reflectances, which are simply the linear averages of the blue asterisks. As in the idealized case shown in Figure 1, the MODIS resolution cloud reflectances fall below the envelope of the sub-pixel values in both 1-D (Figures 11a and 11b) and 3-D (Figures 11c and 11d) simulations. As a result, retrievals of re,2.1 at MODIS resolution are around 25 μm, substantially larger than the sub-pixel mean value of 19μm, regardless of whether 1-D (Figure 11a) or 3-D (Figure 11c) radiative transfer is used. Retrievals using 3.7 μm reflectance are less strongly affected: MODIS resolution re,3.7 retrievals (20.5 μm and 21 μm, for 1-D and 3-D radiative transfer simulations, respectively) are much closer to the sub-pixel mean.
 The example in Figure 11shows the existence of plane-parallelrebias in our simulation, but is it typical? To investigate this question, we first constructed a sub-pixel variability index for bothτ and rebased on the retrievals from 3-D radiative transfer simulations at LES resolution. The index is defined as the ratio of standard deviation to mean of sub-pixel values. Within the context ofFigure 11, a large sub-pixel variability index indicates that the sub-pixel level points are more scattered in the Nakajima-King LUT, whereas small sub-pixel variability index indicates that the sub-pixel level points are clustered around certain constant contour lines ofτ or re. Figure 12bshows the histograms of the sub-pixel variabilities ofτ and refor the ATEX-clean case (black line). Evidently,τhas much stronger sub-pixel variability thanre. This suggests that the example in Figure 11 is a typical case and the re heterogeneity factor is an important factor in causing the Δre,3.7−2.1. It is worth mentioning that we have also investigated the sub-pixel variability ofτ and re in real MODIS observation. We developed a research level algorithm to retrieve τ or re at 500 m using 500 m radiance observations from the MODIS 0.86 μm and 2.1 μm bands. This algorithm is applied to a MODIS-Terra granule shown inFigure 12a. The sub-pixel variabilities ofτ or re derived from the 500 m MODIS retrievals (red lines) for this granule are shown in Figure 12b. The good agreement between MODIS observation and LES results confirms that τ tends to vary more strongly than re within a 1 km MODIS pixel.
 We now turn to the 3-D radiative transfer effects. First, the impacts of 3-D radiative transfer effects onre,2.1 and re,3.7 are investigated in Figure 13. In the figure we use the difference between the τretrieval based on 3-D radiative transfer simulation (referred to as “3-Dτ”) and that based on 1-D radiative transfer simulation (referred to as “1-Dτ”) as an index of the 3-D radiative transfer effects. By definition, 3-Dτ> 1-Dτfor the illuminating effect, and 3-Dτ< 1-Dτ for shadowing. As expected, the reretrievals based on 3-D radiative transfer simulation (“3-Dre”) appear smaller than those based on 1-D radiative transfer simulation (“1-Dre”) for the illuminating effect (i.e., 3-Dre− 1-Dre< 0), and vice versa for shadowing (i.e., 3-Dre− 1-Dre > 0). What is interesting in Figure 13is that the impact of 3-D radiative effects onre,2.1 is generally greater than for re,3.7 . This is likely attributable to the stronger absorption in the 3.7 μm band that acts to reduce the horizontal transport of photons. The impact of 3-D radiative transfer effects on Δre,3.7−2.1 is investigated in Figure 14 as a function of SZA and retrieval resolution. At SZA = 50°, illuminating and shadowing effects are clearly seen in both LES (Figure 14a) and MODIS (Figure 14b) resolution retrievals. Interestingly, based on the 3-D radiative transfer simulations, Δre,3.7−2.1 > 0 over the illuminated side, and Δre,3.7−2.1 < 0 over the shadowing side. These are consistent with the result in Figure 13 that illuminating and shadowing effects have stronger impacts on re,2.1 than on re,3.7. For example, over the illuminating side the apparent brightening decreases the re retrieval, and this impact is stronger for re,2.1 than re,3.7, leading to positive Δre,3.7−2.1. This also explains the origin of the positive Δre,3.7−2.1 points in Figures 7 and 8. At SZA = 20°, a significant portion of 3-Dτretrievals are smaller than their 1-D counterparts. This is likely attributable to horizontal photon transport. Unlike illuminating and shadowing effects, which are a result of geometrical cloud top height variation, the horizontal photon transport effect is a result of horizontalτ variation. The absorption in the 2.1 μm and 3.7 μm bands reduces horizontal photon transport, which explains why the absolute values of Δre,3.7−2.1 in Figure 14c are generally smaller than those of the low sun case in Figure 14a.
 Another interesting point to note in Figure 14 is that, at LES resolution (i.e., Figures 14a and 14c), Δre,3.7−2.1 shows no obvious positive or negative bias. However, at MODIS resolution, Δre,3.7−2.1 seems biased more toward negative values, which is consistent with the results in Figures 5 and 6. This shift of Δre,3.7−2.1 is more clearly seen in Figure 15, which shows Δre,3.7−2.1 averaged over all cloudy pixels with τ> 5 in the domain for different solar and zenith angles, as well as different retrieval resolutions. In both low sun (SZA = 50°) and high sun (SZA = 20°) simulations, and for all viewing zenith angles, the domain-averaged MODIS resolution Δre,3.7−2.1(red lines) based on 3-D radiative transfer simulations is systematically smaller than its LES resolution counterpart (black lines). This indicates that the shift of Δre,3.7−2.1from close-to-zero values to more negative values when reflectance is averaged from high resolution to MODIS resolution is a robust result, only weakly affected by solar and viewing zenith angles.
 The results in Figures 14 and 15seem to suggest that although the 3-D radiative transfer effect has a strong impact onre retrievals, it tends to result in random errors rather than systematic bias. Therefore it is reasonable to hypothesize that the systematic shift of Δre,3.7−2.1 seen in Figures 6 and 15is mainly attributable to the plane-parallelre bias. This hypothesis is further investigated in Figures 16 and 17. Figure 16shows a comparison of the MODIS resolution retrievals with the mean of the LES resolution retrievals (referred to as the sub-pixel mean hereafter). In the case of theτretrieval, the plane-parallel bias is evident: values retrieved from the averaged radiances (i.e., MODIS resolution retrieval) are significantly smaller than the sub-pixel mean values. However, in the case ofre,2.1retrievals, the reverse is true: MODIS resolution retrievals tend to be larger than the sub-pixel mean values. This difference indicates that in comparison with other factors, such drizzle and 3-D radiative transfer, the plane-parallelre bias is the dominant source of errors in the re,2.1retrievals at MODIS resolution. The relationship between the plane-parallelre bias and Δre,3.7−2.1 is shown more clearly in Figure 17. In both 1-D and 3-D simulation, the magnitude of Δre,3.7−2.1is, in general, positively correlated with the strength of the plane-parallelrebias because the plane-parallelre bias has a stronger impact on the re,2.1 than on re,3.7. As this bias increases, re,2.1deviates from the sub-pixel mean, whilere,3.7remains relatively close to the sub-pixel mean, thereby resulting in negative and increasing Δre,3.7−2.1. Together Figures 16 and 17support the hypothesis that the plane-parallelre bias is the primary reason for the systematic bias in the re,2.1 retrievals, leading to significant Δre,3.7−2.1 values. It should be mentioned here that Marshak et al.  found that re,2.1 retrievals tend to decrease with increasing horizontal scale, which is opposite to what we found here. The difference between these two studies might be attributable to the use of different LES fields, different levels of complexity in cloud microphysics (a constant re = 10 μm for the whole cloud field was used by Marshak et al. , while more realistic re fields from a bin microphysics scheme are used in this study), or different configurations in radiative transfer simulations.
 To summarize the lessons learned from the above case studies, we use the following equation to qualitatively describe the impact of various factors on MODIS re, retrieval:
where the subscript λ indicates the spectral band (i.e., 2.1 μm or 3.7 μm) used in the retrieval. In this equation, we use the r*e,λ term to denote the “ground truth,” the Δre,λ3Dterm to represent the impact of the 3-D radiative effects (i.e., illuminating or shadowing effects) onre retrieval and the Δre,λPPterm for the plane-parallelre bias. First, it is important to note that even the “ground truth” r*e,λ is dependent on the spectral band owing to, for instance, the spectral difference in vertical weighting. Second, our case studies demonstrate that the Δre,λ3D and Δre,λPP terms and their relative impact on re,λ depend on both spectral band and spatial resolution. At high resolution (i.e., LES resolution), the Δre,λPP term is small and Δre,λ3Dterm dominates, leading to substantial difference between 1-D and 3-D retrievals (seeFigure 13). Note that the sign of Δre,λ3Dterm could be either positive or negative, depending on the nature of the 3-D effect (i.e., illuminating or shadowing). From a spectral perspective, because the 3-D effects tend to impactre,2.1 more than re,3.7, the absolute value of Δre,2.13D tends to be greater than Δre,3.73D (see Figure 14). At high resolution the sign of Δre,3.7−2.1 could be either positive or negative (see Figure 5). When cloud reflectance is aggregated to MODIS resolution, the cancellation of opposing 3-D effects reduces the absolute value of the Δre,λ3D term. At the same time, the Δre,λPPterm from the plane-parallelrebias becomes significant, especially for heterogeneous pixels. Note that because the sub-pixel scale variability ofre is significantly smaller than that of τ (see Figure 12), the sign of the Δre,λPP term tends to be positive (see Figure 1). From a spectral perspective, because the 3.7 μm look-up-table is more orthogonal than 2.1μm look-up-table, Δre,2.1PP tends to be larger than Δre,3.7PP (see Figures 1 and 11). As a result, at MODIS resolution the Δre,3.7−2.1 is biased toward negative values (see Figures 6 and 15) and generally decreases with increasing sub-pixel heterogeneity (seeFigure 7), although for individual pixels Δre,3.7−2.1 could be either positive or negative depending on the Δre,λ3D term. Finally, it should be noted that the Δre,λ3D and Δre,λPPterms arise only when a cloud field has significant horizontal heterogeneity. Therefore, these terms cannot explain why some very homogenous pixels in MODIS operational retrievals (i.e., upper-left part ofFigure 2) have large negative values of Δre,3.7−2.1 (about −5 μm). An explanation for such pixels is left for future studies.
6. Conclusions and Future Work
 In this study, we develop a MODIS cloud property retrieval simulator based on the combination of a large-eddy simulation model and radiative transfer models. Using this simulator, MODIS retrievals of shortwave optical depth (τ) and droplet effective radii using 2.1 or 3.7 μm radiances (re,2.1 and re,3.7) are simulated at different resolutions based on both 1-D and 3-D radiative transfer for several LES cases. The effects of drizzle and cloud horizontal inhomogeneity on thereretrievals in these cases are investigated. It is found that at high resolution (∼100 m) 3-D radiative transfer effects, such as enhanced illumination and shadowing, affectre,2.1 stronger than re,3.7 probably attributable to weaker absorption at 2.1 μm. As a result, the illumination effect tends to result in positive Δre,3.7−2.1 (= re,3.7 − re,2.1) and the shadowing effect tends to result in negative Δre,3.7−2.1. However, because of a balance between these two opposing effects, re,2.1 and re,3.7agree reasonably well at high resolution, with no systematic bias between the two. At MODIS-like resolution (∼800 m),re,2.1 is found to become systematically larger than re,3.7 and the difference is seen to increase with re,2.1and the sub-pixel inhomogeneity indexHσ, consistent with the trends found in operational MODIS observations. This difference is unlikely a direct radiative result of drizzle here because the removal of drizzle in the radiative computations has little impact on the retrievals, including Δre,3.7−2.1. It is also found that opposing 3-D radiative effects tend to cancel each other out at MODIS-like resolution, resulting in a weaker net impact of 3-D effects onreretrievals. Finally, it is found within pixels of MODIS-like resolution that cloudτ generally varies more strongly than re. The strong sub-pixel variability inτgives rise to an plane-parallelrebias that results from averaging over non-orthogonalre and τretrieval curves in the 2-channel (visible and SWIR) radiance phase space, and this heterogeneity bias is largely responsible for the systematic difference betweenre,3.7 and re,2.1at MODIS-like resolution
 Our results have several implications. First, it is evident from the our analysis, as well as previous studies [Boeke, 2009; Hayes et al., 2010; Seethala and Horváth, 2010; Zhang and Platnick, 2011], that the cloud horizontal inhomogeneity effect can cause substantial errors in MODIS re,2.1 retrievals. The MODIS re retrieval based on the 1.6 μm band (re,1.6) is not considered in this study but, given that water absorption is even weaker in this band, we expect that the re,1.6 retrieval faces the same problem. Further investigations are needed to understand the temporal and geographical distributions of this error in the MODIS cloud product, and its implications for climate and aerosol indirect effect studies. Second, it is shown that the inhomogeneity index Hσ can be used to assess the magnitude of such errors, and therefore the quality of MODIS operational re,2.1 retrievals. In fact, in the upcoming MODIS Collection 6 (C6) cloud product, it has been planned for the index Hσ to be reported as part of the cloud mask product (MOD35). Finally, the large negative Δre,3.7−2.1 values associated with large Hσ index (i.e., points in the upper right corner of Figure 2) are likely the result of the cloud inhomogeneity effect. This finding warrants caution for the use of these retrievals in cloud vertical structure or drizzle detection algorithms based on Δre,3.7−2.1 [Chang and Li, 2002, 2003; Kokhanovsky and Rozanov, 2011].
 Several questions also arise from this study that are worthy of further investigation. For instance, is it possible to reduce the impact of cloud inhomogeneity on the re,2.1 retrieval? Note that the MODIS 2.1 μm band has the native resolution of 500 m. For the limited cases in this study, re,3.7 and re,2.1 agree better at higher resolution. Therefore, it is worth exploring whether the magnitude of Δre,3.7−2.1is substantially reduced if retrievals are made at 500 m resolution. Such investigations would also provide useful information for other satellite instruments, for instance the VIIRS (Visible/Infrared Imager Radiometer Suite) on the Suomi NPP (National Polar-orbiting Partnership) mission, which has a resolution of about 375 m. Another intriguing question is whether Δre,3.7−2.1 contains any useful information for cloud vertical structure retrieval or drizzle detection. Note that a significant portion of the large negative Δre,3.7−2.1 values are found to be associated with small Hσ index in the MODIS observations (i.e., the pixels in the upper left of Figure 2). Such situations are not represented in our simulation cases. It seems that the cloud inhomogeneity effect is small for such pixels. It remains unclear and warrants further investigation whether these large negative Δre,3.7−2.1 values associated with small Hσ reflect the cloud vertical structure [Chang and Li, 2002, 2003; Kokhanovsky and Rozanov, 2011] or other factors. Finally, future work is also needed to study the impact of cloud horizontal inhomogeneity on ice cloud microphysics retrievals and on other remote sensing techniques, for example the infrared split window method [Inoue, 1985; Prabhakara et al., 1988].
 We thank Daniel Grosvenor and the other two anonymous reviewers for their insightful comments, questions, and suggestions, which have helped to improve this manuscript. ZZ, AA and SP were supported by NASA under grant NNX11AI98G, and RP was supported by NASA under grant NNX11AF09G. Computational support was provided by the NASA Advanced Supercomputing Division.