The Moderate Resolution Imaging Spectroradiometer (MODIS) cloud product provides three separate 1 km resolution retrievals of cloud particle effective radii (re), derived from 1.6, 2.1 and 3.7 μm band observations. In this study, differences among the three size retrievals for maritime water clouds (designated as re,1.6re,2.1 and re,3.7) were systematically investigated through a series of case studies and global analyses. Substantial differences are found between re,3.7 and re,2.1 retrievals (Δre,3.7–2.1), with a strong dependence on cloud regime. The differences are typically small, within ±2 μm, over relatively spatially homogeneous costal stratocumulus cloud regions. However, for trade wind cumulus regimes, re,3.7 was found to be substantially smaller than re,2.1, sometimes by more than 10 μm. The correlation of Δre,3.7–2.1 with key cloud parameters, including the cloud optical thickness (τ), re and a cloud horizontal heterogeneity index (Hσ) derived from 250m resolution MODIS 0.86 μm band observations, were investigated using one month of MODIS Terra data. It was found that differences among the three re retrievals for optically thin clouds (τ < 5) are highly variable, ranging from −15 μm to 10 μm, likely due to the large MODIS retrieval uncertainties when the cloud is thin. The Δre,3.7–2.1 exhibited a threshold-like dependence on both re,2.1 and Hσ. The re,3.7 is found to agree reasonably well with re,2.1 when re,2.1 is smaller than about 15 μm, but becomes increasingly smaller than re,2.1 once re,2.1 exceeds this size. All three re retrievals showed little dependence when cloud is relatively homogenous (Hσ < 0.3 defined as standard deviation divided by the mean for the 250 m pixels within a 1 km pixel retrieval). However, for inhomogeneous clouds (Hσ > 0.3), both re,1.6 and re,2.1 were seen to increase quickly with Hσ. On the other hand, re,3.7 statistics showed little dependence on Hσ and remained relatively stable over the whole range of Hσ values. Potential contributing causes to the substantial re,3.7 and re,2.1 differences are discussed. In particular, based on both 1-D and 3-D radiative transfer simulations, we have elucidated mechanisms by which cloud heterogeneity and 3-D radiative effects can cause large differences between re,3.7 and re,2.1 retrievals for highly inhomogeneous clouds. Our results suggest that the contrast in observed Δre,3.7–2.1 between cloud regimes is correlated with increases in both cloud re and Hσ. We also speculate that in some highly inhomogeneous drizzling clouds, vertical structure induced by drizzle and 3-D radiative effects might operate together to cause dramatic differences between re,3.7 and re,2.1 retrievals.
 Low-level maritime water clouds play a critical role in both global and regional climate, through their strong radiative effects [Klein and Hartmann, 1993] and interactions with aerosols and precipitation [e.g., Haywood and Boucher, 2000; Lohmann and Feichter, 2005]. Among many others, the cloud particle effective radius (re) is perhaps the most important cloud microphysical parameter. It not only has a significant influence on cloud radiative forcing responses to aerosol modification [Oreopoulos and Platnick, 2008], but is also a key proxy useful for understanding aerosol-cloud-precipitation interactions [Twomey, 2007; Albrecht, 1989; Lohmann and Feichter, 2005]. The importance of re has motivated the development of various satellite-based remote-sensing techniques for monitoring the global cloud re from space [e.g., Prabhakara et al., 1988; Platnick and Twomey, 1994; Breon and Doutriaux-Boucher, 2005]. Of particular interest in this study are cloud re retrievals from the MODerate Resolution Imaging Spectroradiometer (MODIS). The operational MODIS cloud retrieval algorithm (product names MOD06 and MYD06 for MODIS Terra and Aqua, respectively) fundamentally utilizes cloud reflectance measurements from two MODIS bands, typically one in a non-absorbing spectral region and the other from a shortwave infrared (center wavelengths near 1.6 and 2.1 μm) or midwave infrared (3.7 μm) band, to retrieve cloud optical thickness (τ) and cloud re simultaneously [Nakajima and King, 1990]. In the MODIS operational cloud product (MOD06) Collection 5 (C5) processing algorithm, the combination of the 0.86 and 2.1 μm bands are used for τ and re retrievals over open ocean [Platnick et al., 2003]. Hereafter, we will refer to the re retrieval based on the 2.1 μm band observation as re,2.1. Besides re,2.1, the MODIS also provides two other re retrievals, one based on the 1.6 μm band and the other based on the 3.7 μm band observations (hereafter referred to as re,1.6 and re,3.7).
 Because MODIS offers three separate re retrievals, several questions naturally arise, namely: How do the three microphysical retrievals differ statistically from each other? If and when/where they differ, can various candidate causes be diagnosed or de-convolved? These are important questions for a number of reasons. First, if differences can be explained in terms of cloud properties that are not directly retrieved by current passive imager algorithms as hypothesized in some previous studies (e.g., cloud vertical structure [e.g., Platnick, 2000] or the presence of drizzle [e.g., Chang and Li, 2002; Nakajima et al., 2010b], then those differences convey potentially useful information about these parameters. Second, an understanding of the differences may help identify algorithm artifacts and therefore improve current passive retrieval algorithms. Third, the answers will provide insights into differences between MODIS re retrievals and those from other satellite instruments that use either a similar remote sensing technique, such as the AVHRR (Advanced Very High Resolution Radiometer) or SEVIRI (Spinning Enhanced Visible and Infra-Red Imager), or a different technique, such as POLDER (Polarization and Directionality of the Earth's Reflectance). Finally, answers to the above questions will provide some guidance for the use of MODIS re retrievals in studies involving cloud processes, aerosol indirect effects, cloud climatologies, and parameterizations in climate models.
 The importance and potential implications of spectral retrieval discrepancies has motivated a number of previous studies, most being theoretical in nature and based on synthetic data. For example, using so-called “weighting function” approximations, Platnick  quantified the re,3.7 retrieval sensitivity to upper cloud microphysics relative to re,2.1 and re,1.6 retrievals that penetrate deeper. An important implication from Platnick  is that if cloud re increases monotonically from cloud base to cloud top, as in a classic adiabatic growth model [Brenguier et al., 2000], re,3.7 should be larger than re,2.1, which in turn is somewhat larger than re,1.6, with a magnitude that depends on cloud optical thickness, microphysical details, and geometry. Based on the theoretical ideas of Platnick , Chang and Li [2002, 2003] proposed that warm rain process (i.e., collision-coalescence) can lead to decrease-with-height re structure at cloud top, which in turn leads to smaller re,3.7 retrievals than re,2.1 due to their different sensitivity to re vertical structure. The potential impacts of drizzle and warm rain processes on MODIS re retrievals were investigated in recent studies by Nakajima et al. [2010b, 2010a] and Suzuki et al.  using collocated MODIS and CloudSat radar observations. In addition to adiabatic growth and warm rain processes, evaporation due to cloud top entrainment [Breon and Doutriaux-Boucher, 2005; Seethala and Horváth, 2010] and 3-D radiative effects caused by cloud horizontal heterogeneity [e.g., Marshak et al., 2006; Boeke, 2009; Hayes et al., 2010; Wolters et al., 2010] have also been suggested to play a role in causing differences between MODIS re,3.7 and re,2.1 for marine water clouds.
 A systematic assessment of differences from the operational MODIS cloud product has been lacking until recently. In a comparison study of global cloud water path from MODIS and AMSR-E (Advanced Microwave Scanning Radiometer – EOS), Seethala and Horváth  noted that MODIS re,3.7 retrievals for non-precipitating marine water clouds can be significantly smaller than re,1.6, and the difference shows a strong dependence on cloud regime. They suggested that cloud top entrainment and 3-D radiative effects are potential reasons for the observed differences. Similarly, Nakajima et al. [2010b] also found that MODIS re,3.7 can be significantly smaller than its re,1.6 and re,2.1 counterparts for marine water clouds. Based on analysis of collocated CloudSat and MODIS observations, Nakajima et al. [2010a] and Suzuki et al.  suggested that the microphysical structure caused by warm rain processes might play an important role in causing the differences. These recent observational studies have shed light on MODIS re spectral retrieval differences for marine water clouds. However, the studies have been inherently limited by their sampling techniques. For example, in the work by Seethala and Horváth  the comparison between MODIS and AMSR-E products focused mainly on the AMSR-E overcast scenes (0.25° resolution). In work by Nakajima et al. [2010a] a cloud must be seen by both the larger CloudSat footprint as well as MODIS to be sampled, and it is known that CloudSat misses a significant portion of marine water clouds due to sensitivity and surface clutter issues [Marchand et al., 2008].
 A final sampling issue to note is that, although the three re retrievals are based on the same algorithm, each MODIS size retrieval has a different failure rate due to different sensitivities to, for example, instrument accuracy, ancillary data, and, as will be shown later, 3-D effects. In the MODIS Collection 5 (C5) cloud product, re,1.6 and re,3.7 are reported as their respective differences from successful re,2.1 retrievals. Thus, the sampling of re,1.6 and re,3.7 are biased by the success of re,2.1 retrieval. The potential impacts of this sampling bias on the statistics of re,1.6 and re,3.7 could not be addressed in previous studies and therefore remain unknown. In the upcoming MODIS cloud product Collection 6 (C6), it is planned for the three re retrievals to be reported independently in both the Level-2 and Level-3 products. This may provide a new opportunity for better understanding the microphysics of marine water clouds.
 We note that the two general mechanisms discussed in previous studies (drizzle and droplet vertical/horizontal inhomogeneities) may be correlated in many regimes, and no study has had the obvious means to untangle such relationships. Spectral retrieval differences may reflect the combined influence of these causes and/or others, and therefore, attributing re differences solely to a single cloud parameter or microphysical process may be misleading. It is important to have an inclusive understanding of re retrieval differences.
 Motivated by the above considerations, we have performed a global assessment of the differences among MODIS re,1.6., re,2.1 and re,3.7 for marine water clouds. In particular, we have analyzed the correlations of re differences with several key cloud parameters, as well as other important factors in the MODIS cloud retrieval algorithm. In the sections that follow, we present the major findings in the context of previous studies and discuss potential implications. The primary goal is to document the differences among the MODIS re,1.6., re,2.1 and re,3.7 for marine water clouds, and the dependence of these differences on key cloud parameters. In section 2.1, the data used in this study is described. In section 2.2 we analyze the re retrievals for two MODIS data granules. Results from an analysis of global statistics is reported in section 2.3. In section 3, we revisit two hypotheses proposed by previous studies to explain MODIS re differences and provide some new insights based the results from section 2. Finally, the main findings are summarized in section 4.
2. Observational Study
 The MODIS data used for this study are from a research-level version of the Collection 5 MODIS cloud product (MOD06). Detailed descriptions of the 1 km operational algorithm is provided by King et al. , Platnick et al.  and Hubanks et al. . As mentioned, three separated re retrievals, based on the MODIS 1.6 μm, 2.1 μm and 3.7 μm band observations, are reported in the operational MOD06 cloud product; it is important to note, however, that only the re,2.1 retrieval is explicitly reported, whereas re,1.6 and re,3.7 are reported as differences with respect to re,2.1 (i.e., Δre,1.6–2.1 and Δre,3.7–2.1). Thus the sampling of re,1.6 and re,3.7 is restricted by the success of the re,2.1 retrieval. In contrast, a research algorithm has been modified such that this condition is relaxed and all three re retrievals are explicitly reported. The impact of the conditional sampling scheme used in the C5 product on the statistics of re,1.6 and re,3.7 is subsequently investigated. In the following analyses, only those 1-km MODIS pixels that meet the following conditions are selected: 1) labeled as “confident cloudy” by the 1 km MODIS cloud mask product (MOD35 [Ackerman et al., 1998]); 2) over ocean; 3) labeled as “liquid water” by the MODIS 1-km “Cloud_Phase_Optical_Properties” data set within MOD06; 4) cloud top temperature warmer than 273K. These conditions are expected to remove most of the situations that may complicate the analyses, such as thin cirrus overlapping lower clouds. All data used in this study including those from our research-level algorithm are available upon request.
2.2. Case Study
 We begin our analysis with MODIS Level-2 (pixel-level) data, which unlike Level-3 gridded data, provides information unaffected by spatial and temporal averaging. Figure 1 shows the true-color image of a Terra MODIS data granule collected on April 2nd, 2005 over the southeast Pacific Ocean (35S∼15S; 100W∼75W). An interesting feature of this granule is the obvious east-to-west transition from closed-cell stratocumulus to broken cumulus. Such transitions are thought to result from a combined effect of increasing marine boundary layer depth, decreasing lower-tropospheric stability and CCN, and increasing drizzle from the coastal to open ocean region [Wood and Hartmann, 2006; Feingold et al., 2010].
 The τ and re,2.1 retrievals from the research-level algorithm are shown in Figures 2a and 2b, respectively. Here, τ remains relatively constant and, if anything, decreases slightly from the closed-cell to open-cell region, while re,2.1 increases from about 10 μm in the close-cell region to about 25 μm in the open-cell region. The re retrieval differences, Δre,1.6–2.1 and Δre,3.7–2.1, are shown in Figures 2c and 2d, respectively. At first glance, Figures 2b and 2d seem to suggest a negative correlation between Δre,3.7–2.1 and re,2.1 (i.e., Δre,3.7–2.1 becomes more negative with the increasing re,2.1). In the open-cell region, the difference can be as large as −5 μm. In comparison, the Δre,1.6–2.1 remains relatively small (within about ±2 μm) over the whole granule (see Figure 2c). The probability distribution functions (PDFs) of the three re retrievals and their differences are plotted in Figures 2e and 2f, respectively. Evidently, the PDFs of re,1.6 and re,2.1 are quite similar, while the PDF of re,3.7 is somewhat narrower. The re,3.7 retrieval almost never exceeds 25 μm, while re,1.6 and re,2.1 retrievals are larger than 25 μm for about 10% of the selected pixels. The PDF of Δre,1.6–2.1 (the blue line in Figure 2f) is close to Gaussian, which seems to suggest that for this granule the difference between the two is caused mainly by random uncertainties (e.g., instrument noise and ancillary data uncertainty). The PDF of Δre,3.7–2.1 (red line in Figure 2f), however, is substantially negatively skewed. About 80% of the selected pixels in this granule have Δre,3.7–2.1 less than zero, with 25% less than −2 μm, which is on the same order as the re,2.1 uncertainty caused by 15% error in 2.1 μm cloud reflection measurement assuming τ > 5 and re = 15 μm [Platnick and Valero, 1995; King et al., 1997].
 As previously mentioned, mechanisms such as warm rain process (i.e., drizzle) and 3-D radiative effects have been proposed in previous studies as potential causes of substantial re,3.7 and re,2.1 differences, similar to the open cell results shown in Figure 2. While it is difficult to address the issues related to drizzle with MODIS observations alone, the fine resolution (250 m) of the MODIS 0.86 μm band offers an opportunity to explore the correlation between Δre,3.7–2.1 and cloud horizontal inhomogeneity. Recently, it was shown by Liang et al.  and Di Girolamo et al.  that a simple cloud horizontal heterogeneity index, derived from 250m resolution MODIS 0.86 μm band cloud reflectance observations, can be used as a proxy to predict the magnitude of MODIS retrieved optical thickness view angle differences (with a single view, plane-parallel assumption) from Multiangle Imaging SpectroRadiometer (MISR) observations. This index is defined as follows [Liang et al., 2009]:
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 250m-resolution sub-pixels within the 1 km retrieval footprint. Thus, Hσ has a spatial resolution (i.e., 1 km) consistent with the cloud property retrieval and increases with pixel inhomogeneity.
Figure 3 shows Hσ calculated for the MODIS granule in Figure 1. A transition is evident, from relatively small values (i.e., log10(Hσ) < −1.5, or Hσ < 0.03) in the closed-cell region over the eastern part of the granule to relatively large values (i.e., log10(Hσ) > 0.5, or Hσ > 0.3) in the broken cumulus region over the western part of the granule. A side-by-side comparison of Figure 2d and Figure 3a indicates that the Δre,3.7–2.1 tends to be close to zero when the cloud is relatively homogenous (i.e., small Hσ) and becomes increasingly negative as the cloud becomes more inhomogeneous (increasing Hσ). This point can be seen more clearly in Figures 3b and 3c, in which the black, blue and red lines indicate the PDF of Δre (either Δre,1.6–2.1 or Δre,3.7–2.1) for all pixels, and the most homogenous (Hσ < 0.03) and inhomogeneous (Hσ > 0.3) pixels, respectively. It is evident that the PDF of Δre,3.7–2.1 becomes more negatively skewed as Hσ increases, i.e., that re,3.7 becomes increasingly smaller than re,2.1 as sub-pixel cloud inhomogeneity increases. The Δre,1.6–2.1 also shows dependence on Hσ, although to a much lesser extent. It is seen that the PDF of Δre,1.6–2.1 for the most inhomogeneous pixels is broader and shifted slightly to the positive values in comparison with that based on the most homogenous pixels.
Figure 4 shows the true-color image for another Terra MODIS granule over the central Pacific Ocean. The middle of this granule is mostly covered by broken cumulus clouds, and the northeast and southwest corners are covered by ice clouds. An analysis of the MODIS cloud property retrievals, especially of the differences between the three re, is shown in Figure 5. Perhaps the most striking feature in Figure 5 is that, most of the cumulus clouds in the center region of the granule are associated with “missing” or non-retrieved pixels (i.e., gray color). While most of the cumulus clouds in this granule are successfully identified by the MODIS cloud mask algorithm (i.e., MOD35), they are absent from Figure 5 largely due to two reasons. First, a large number of cloud pixels in this granule, especially those broken cumulus, are restored to clear-sky pixels during the “Clear-Sky Restoral” (CSR) algorithm within the MOD06 optical retrieval algorithm. In this instance, the CSR algorithm is mostly eliminating cloud obstructed fields of view that are on the edge of clear regions, but also uses MOD35 250 m cloud detection inhomogeneity to eliminate partly cloudy pixels from consideration. Second, only those pixels in which all three re retrievals are successful have been plotted. Because of this conditional sampling scheme, a significant number of pixels that survived the CSR algorithm, but resulted in at least one failed re retrieval, are plotted as “missing” (a detailed analysis of these “missing” pixels is discussed below).
 The PDFs of the re,1.6, re,2.1 and re,3.7 for those pixels in which all three re retrievals were successful are shown in Figure 5e. The PDFs of re differences are shown Figure 5f. For the water clouds in this granule, re,3.7 is substantially smaller than both re,1.6 and re,2.1 . The mean values of re,1.6 and re,2.1 are both around 20 μm, while the mean value of re,3.7 is only about 14 μm. Figure 5f also indicates that almost all water cloud pixels in this granule have smaller re,3.7 than re,2.1, and for half of the pixels re,3.7 is more than 5 μm smaller.
Figure 6 shows the sub-pixel heterogeneity index Hσ for this broken cloud case. As expected, the broken cumulus cloud pixels in the central region of this granule have quite large Hσ, indicating that these clouds have large horizontal heterogeneity. From Figure 6, one can also note that the Δre,1.6–2.1 shows little dependence on Hσ. The Δre,3.7–2.1, on the other hand, clearly increases with increasing Hσ, which suggests the potential role of cloud horizontal heterogeneity.
 The broken cumulus clouds in Figure 4 are extremely challenging for the MODIS cloud retrieval algorithm. The plane-parallel assumption, on which the MODIS retrieval algorithm is based, does not hold for these clouds. Various 3-D radiative effects, such as shadowing, illuminating and horizontal photon transport, make it difficult to connect MODIS observations with cloud physics (i.e., τ and re). The CSR algorithm in the current MODIS operational retrieval algorithm is designed to identify those cloudy pixels that are highly inhomogeneous and/or likely to be non-ideal (i.e., partly cloudy). Figure 7a shows the CSR result for this granule. Gray indicates clear-sky pixels as determined by the cloud mask (MOD35), while yellow indicates pixels that are identified by MOD35 as having cloudy fields of view but are restored to “clear-sky” before optical retrievals are performed (i.e., no τ and re retrievals are attempted). It is not surprising to see that a large number of pixels in this granule are labeled as CSR due to the significant number of clear-cloudy edge pixels (not analyzed). To investigate the impact of CSR on the statistics of the three re retrievals, we re-processed this granule with the CSR algorithm turned off. The results are shown in Figure 7b and Table 1. Most of the CSR pixels in Figure 7a (yellow) become failed pixels (blue), i.e., re,2.1 and/or re,3.7 failed in Figure 7b when CSR is turned off. Quantitatively, we found that by turning off CSR a total of 466,587 additional water cloud pixels (about 95% of all additionally gained pixels) are gained. However, only about 16% and 23% of these pixels result in successful re,1.6 and re,2.1 retrievals, respectively. A significant number of CSR pixels, about 60%, result in successful re,3.7 retrievals. This would appear to suggests that re,3.7 is considerably less affected by cloud horizontal heterogeneity and 3-D radiative effects than both re,1.6 and re,2.1 and/or there are offsetting biases due to thermal emission corrections. What is especially interesting is that the additional pixel counts gained by turning off the CSR algorithm appear to have even larger Δre,3.7–2.1 (see “All vs. Additional” in Figure 8, bottom). This is not surprising though because the CSR pixels generally have larger Hσ and therefore more susceptible to 3-D effect. The analysis of the CSR test in Figure 8 indicates that most of the CSR-identified pixels would result in failed re retrievals anyway, and those that are successful tend to result in large Δre. These results justify the considerations behind the CSR test. In the Collection 5 data set, CSR pixels are given the lowest Quality Assurance (QA) value and thereby not aggregated to Level-3.
Table 1. For the Granule Shown in Figure 4, the Number of Additional Water Cloud Pixels Retrieved When Clear-Sky Restoral Tests Are Turned Off in the MOD06 Algorithm and the Resulting Retrieval Statistics for These Additional Pixelsa
Number of Additional Pixels Retrieved
Percent of Additional Successful re Retrieval With Reference to Total Additional Water Cloud Pixels
Mean Value of Additional re Retrieval
Note that when CSR is turned on a total of 245901 pixels are determined as water-phase. When CSR test is turned off for this granule a total of 466587 additional water cloud pixels are gained.
Figure 8 also shows the impacts of CSR and conditional sampling on the statistics of the three re retrievals for the granule in Figure 7. Figure 8 (top) shows the PDFs of re,1.6, re,2.1 and re,3.7 retrievals for those pixels that pass all three re retrievals when CSR is on. Figure 8 (middle) shows the re PDFs when CSR is on and the three re retrievals are sampled independently. Figure 8 (bottom) shows the re PDFs when CSR is turned off in the retrieval and the three re retrievals are sampled independently. Although the CSR and conditional sampling are seen to have notable impacts on the PDF of re,3.7, these impacts by no means change the conclusion that re,3.7 is substantially smaller than re,1.6 and re,2.1. It seems therefore safe to conclude that the substantial difference between re,3.7 and re,2.1 is robust and unlikely to be the result of algorithm issues such as conditional sampling and CSR.
 The above analysis confirms the findings of several previous studies that the MODIS re,3.7 and re,2.1 retrievals for marine water clouds can be substantially different. The most important lesson learned from the analysis is perhaps that the difference between re,3.7 and re,2.1 is correlated with increasing re and increasing cloud heterogeneity. This feature of the re retrieval difference will be further explored in the global study presented in the next section. The case studies also suggest that the difference between MODIS re,3.7 and re,2.1 is unlikely to be a result of MOD06 algorithm choices, such as conditional sampling and CSR, but due to more fundamental reasons.
2.3. Global Study
 In this section, we explore the differences between re,1.6, re,2.1 and re,3.7 on a global scale. The analysis is based on one month of Terra MODIS Level-3 data from April 2005. Because re,1.6 and re,3.7 are not currently aggregated in the Collection 5 Level-3 MODIS cloud product (MOD08), we have directly aggregated re,1.6 and re,3.7 from the Level-2 1 km data to a 1° Level-3 in a manner consistent with the operational aggregation of re,2.1.
2.3.1. Dependence on Cloud Regime
 The monthly mean re,1.6, re,2.1 and re,3.7 for marine water clouds for April 2005 are shown in Figures 9a, 9b and 9c, respectively. The water cloud fraction from successful MODIS cloud optical property retrievals (i.e., “Cloud_Fraction_Liquid_FMean” in MODIS Level-3 product retrievals of both τ and re are consistent with look up table physics and ancillary data) is shown in Figure 9d. Several well-known coastal stratocumulus regions, for example off the coasts of California, Peru and Namibia/Angola, are clearly seen in Figure 9. One can also see from Figure 9 that the transition from the costal stratocumulus cloud regimes, with water cloud fraction > 60%, to the offshore cumulus cloud regimes, with water cloud fraction < 20%, is quite sharp. Accompanying this transition, re,1.6 and re,2.1 increases substantially from 8 ∼ 10 μm near the coast to as large as 25∼30 μm far offshore. The re,1.6 and re,2.1 means show the same relative spatial distribution and, to some extent, magnitude. However, re,3.7 is significantly smaller than its counterparts, especially over regions with low water cloud fraction. Note that the aforementioned stratocumulus cloud regions and the transition of cloud regime are still clearly visible in Figure 9c, although to a lesser extent.
 For a better comparison, the global distribution of mean Δre,1.6–2.1 and Δre,3.7–2.1 are shown in Figure 10. It is easily seen that Δre,1.6–2.1 and Δre,3.7–2.1 have the opposite sign, indicating that MODIS re,1.6 is generally larger, and re,3.7 is generally smaller, than re,2.1, though by significantly different magnitudes. The Δre,1.6–2.1 is mostly smaller than 2 μm, and lacks a strong regional feature outside of the southern tropics. Conversely, Δre,3.7–2.1 shows an obvious dependence on cloud regime. For example, over coastal stratocumulus cloud regions, Δre,3.7–2.1 is close to zero, or even slightly positive. However, over the broken cumulus cloud regions, where water cloud fraction is small, re,3.7 is seen to be smaller than re,2.1 by as much as 5 μm to 10 μm on average. These features in Figure 9 and Figure 10 are generally in agreement with the findings from the case studies, namely Δre,3.7–2.1 changes from near-zero to large negative values as the cloud regime changes from closed-cell stratocumulus (relatively homogenous) to open-cell cumulus (more inhomogeneous).
2.3.2. Dependence on Key Cloud Parameters
 In the rest of this section, we explore the correlations between MODIS re retrieval differences and several key factors. In doing so, we attempt to identified regimes where the re retrieval differences can be attributed more to cloud physics, such as drizzle, rather than to retrieval uncertainties and artifacts caused by, for example, 3-D radiative effects. Figure 11a shows the joint histogram between re,1.6 and re,2.1 based on about 1.5 billion Level-2 marine water cloud pixels collected during April 2005 by Terra MODIS from 60S to 60N. Evidently, the density of points is highest along the one-to-one line, attesting that pixel-level re,1.6 and re,2.1 modes agree reasonably well. Figure 11b shows the joint histogram between re,3.7 and re,2.1 . Similar to Figure 11a, the density of points is also highest along the on-to-one line. It is interesting to see that the bias between re,3.7 and re,2.1 is quite small when re is smaller than about 12∼13 μm. However, when re is larger than about 15 μm, the histogram distribution is clearly weighted toward the re,2.1 side and the deviation from the one-to-one line increases with increasing re,2.1. Figures 11c and 11d present the same story, but from a different perspective. Figure 11c shows the joint histogram between Δre,1.6–2.1 and re,2.1. The hump-shaped pattern indicates that the histogram has a tail extending to the positive side of Δre,1.6–2.1. Note that the cut-off behavior in Figure 11c is caused by the fact that in the C5 MODIS operational cloud retrieval algorithm, the re retrieval for water clouds is limited to 30 μm. Figure 11d shows the joint histogram between Δre,3.7–2.1 and re,2.1. It is seen that the Δre,3.7–2.1 centers around zero for re,2.1 between 5∼13 μm and quickly becomes negative when re,2.1 becomes larger than 15 μm. For some outliers re,3.7 is smaller than re,2.1 by more than 10 μm. Overall, the results in Figures 11a–11d are encouraging. On the one hand, they show that on the monthly and global scale, the three different MODIS re retrievals for marine water clouds agree reasonably well for a majority of pixels. On the other hand, the results indicate that for a non-negligible portion of pixels, significant differences do exist.
 It is important to note that the sampling issue has a strong impact on the shape of the joint histogram in Figures 11c and 11d. The gray dashed line in Figures 11c and 11d represents the PDF of re,2.1 derived from one month of Level2 data. (April 2005), The PDF peaks at about 12–13 μm, which mostly explains the joint histogram peak near 12–13 μm. Re-normalizing the joint histograms separately for each re,2.1 bin results in the histograms shown in Figures 11e and 11f. In these bin-normalized histograms, the behavior of Δre,3.7–2.1 (or Δre,1.6–2.1) versus re,2.1 in the range of re,2.1 between 20 to 30 μm becomes much clearer, and some interesting features begin to emerge. In Figure 11e the histogram of Δre,3.7–2.1 versus re,2.1 shows increasing spread with increasing re,2.1, while the bias between Δre,1.6–2.1 does not vary much over the entire re,2.1 range. Interestingly, the bin-normalized histogram of Δre,3.7–2.1 versus re,2.1 not only shows an increasing spread with increasing re,2.1, but also a clear systematic transition, in terms of the most-likely observed Δre,3.7–2.1 for a given re,2.1 (i.e., the red area), from near zero values when re,2.1 < 15 μm to larger negative values when re,2.1 > 15 μm. The fact that this threshold-like behavior takes place at re∼15 μm is particularly interesting because re ∼ 15 μm has been suggested to be the threshold for the collision-coalescence process to take place in marine water clouds [Gerber, 1996]. However, further studies are needed to determine whether this is simply a coincidence or due to more fundamental physical reasons.
Figure 12 shows the bin-normalized joint histograms between re retrieval differences and cloud optical thickness (τ). One can easily note that when clouds are optically thin (e.g., τ < 5) both Δre,1.6–2.1 and Δre,3.7–2.1 vary quite remarkably from −15 μm to 10 μm. However, when the cloud becomes sufficiently thick (τ > 5), the statistics of both Δre,1.6–2.1 and Δre,3.7–2.1 become stable and show little dependence on τ. The behaviors of Δre in Figure 12 are consistent with our understanding of the uncertainties of MODIS cloud property retrievals. When a cloud is thin, the signal from the cloud is comparable, or even smaller, than the noises caused by, for example, instrument uncertainties, ancillary data uncertainties, and discretization and interpolation of the look-up-table. As a result, the uncertainty associated with the MODIS re retrievals for thin clouds is large, which explains the remarkable variation of Δre in Figure 12 when τ is smaller than about 5. Caution must therefore be taken when interpreting the difference between the three MODIS re retrievals for clouds with τ smaller than about 5, as many factors other than cloud physics, such as retrievals errors and artifacts, all play a significant role in this regime.
 In the case studies, we noted that the Δre,3.7–2.1 shows a clear dependence on cloud horizontal heterogeneity. This dependence is further investigated in Figure 13. Figures 13a, 13b and 13c show the joint histograms of the sub-pixel cloud inhomogeneity (Hσ) defined in equation (1) versus, re,1.6, re,2.1, and re,3.7, respectively. Figures 13d and 13e show the joint histograms of the Hσ versus Δre,1.6–2.1 and Δre,3.7–2.1, respectively. Optically thin clouds (τ < 5) are excluded from this figure because MODIS retrievals for these clouds are subject to large uncertainties due to the low signal-to-noise ratio. But results are similar if we include thin pixels (not shown). To remove the sampling concerns, all histograms in Figure 13 are bin-normalized as in Figure 11 and Figure 12, but with respect to Hσ (gray dashed line). The most compelling feature in Figure 13 is the sharp transition of re,1.6 and re,2.1, at Hσ around 0.3∼0.5. When Hσ is smaller than 0.3, the most-likely re,1.6 (i.e., red area in Figure 13a) for a given Hσ stays relatively constant, within 10∼15 μm. However, when Hσ exceeds about 0.3, the most-likely re,1.6 value increases dramatically with Hσ. A similar but less pronounced pattern is also observed in the Hσ versus re,2.1 histogram. Interestingly, this is not the case in Figure 13c, where the most-likely value of re,3.7 shows only weak dependence on the sub-pixel inhomogeneity. It is therefore not surprising to see in Figures 13d and 13e the most-likely values of Δre,3.7–2.1 shifting from near-zero to the negative side when Hσ exceeds about 0.3. Figure 13 confirms that the transition of Δre,3.7–2.1 and Δre,1.6–2.1 with cloud regime observed in the case studies is a robust behavior of global MODIS re retrievals for marine water clouds. The potential reasons for such behavior, including the 3-D radiative effect and the effect of drizzle, will be discussed later in Section 3.
 In addition to re, τ and Hσ, we have also investigated the dependence of Δre,3.7–2.1 on other factors such as cloud top temperature, solar zenith angle, satellite viewing angle, scattering angle, etc., none of which shows an impact on Δre,3.7–2.1 as dramatic and clear as re, τ and Hσ . It is worth mentioning that we have implemented a test version of the Cox-Munk [Cox and Munk, 1954] ocean surface reflectance model in MOD06. Based on preliminary results, we found that the impact of Cox-Munk reflectance model on Δre is negligible regardless of τ.
 The observational studies in the last section reveal significant differences between MODIS re,2.1 and re,3.7 retrievals for marine water clouds, and suggest that these differences depend primarily on three key cloud parameters that can be obtained from MODIS: the cloud optical thickness (τ), effective radius (re), and horizontal heterogeneity (Hσ). It is easily understood that the dependence on τ results from the large uncertainty in the MODIS re retrieval when the cloud is thin. In this section, we will turn our attention to the dependence on re and Hσ. As previously mentioned, a number of hypotheses have been proposed in previous studies to explain the large difference between MODIS re,2.1 and re,3.7 retrievals for marine water clouds. These hypotheses largely fall into two categories: 1) the difference is a result of cloud vertical structure induced by microphysical process such as cloud top entrainment [Seethala and Horváth, 2010] and warm rain process [Chang and Li, 2002; Nakajima et al., 2010b, 2010a; Suzuki et al., 2010] (i.e., drizzle), and 2) the difference, at least to a large extent, can be attributed to artifacts caused by the cloud horizontal heterogeneity and 3-D radiative effects [Boeke, 2009; Hayes et al., 2010; Wolters et al., 2010]. The objective of this section is twofold: first, we will put the results from the observational studies in the context of these two hypotheses and elucidate how cloud vertical structure and 3-D effects could lead to the observed difference between re,2.1 and re,3.7 retrievals; second, we will provide some new insights into these two hypotheses. In particular, we will discuss the possibility that the drizzle process and 3-D effects are inherently coupled and operating together to cause the substantial difference between re,2.1 and re,3.7 in some highly heterogeneous clouds.
3.1. In-Cloud Vertical Structure and Drizzle
 Various microphysical processes, such as adiabatic growth, entrainment mixing, collision-coalescence, and sedimentation, all affect cloud vertical structure. The concept of “weighting function” developed by Platnick  provided a convenient framework to assess the sensitivity of MODIS re retrieval to the vertical structure of cloud. An important implication from Platnick  is that if cloud re increases monotonically from cloud base to cloud top, as in a classic adiabatic growth model [Brenguier et al., 2000], re,3.7 should be larger than re,2.1, which in turn is somewhat larger than re,1.6. However, it is important to note that because the adiabatic condensation growth becomes less efficient as the droplet grows larger, the droplet growth rate dre/dz remains relative small over the upper portion of the water cloud [Brenguier et al., 2000]. As a result, the magnitude of Δre induced by adiabatic cloud vertical structure is expected to be small, generally smaller than 2 μm [Platnick, 2000].
 Several previous studies speculated that cloud top entrainment could lead to a decrease-with-height re structure at cloud top, which in turn leads to smaller re,3.7 retrieval than re,2.1 [Breon and Doutriaux-Boucher, 2005; Seethala and Horváth, 2010]. It should be pointed out that the decrease-with-height re structure at cloud top assumed in these studies seems to indicate a homogenous mixing process [Baker et al., 1980]. On the contrary, observational studies have actually found more inhomogeneous mixing cases than homogenous mixing cases [Gerber et al., 2005]. Note that inhomogeneous mixing may cause changes in cloud horizontal structure (i.e., causing “cloud holes” [Gerber et al., 2005]) in additional to its impact on cloud microphysics. Although attempts have been made to understand the impact of microphysical changes on cloud property retrieval [e.g., Boers et al., 2006], the impacts of cloud horizontal structure changes associated with inhomogeneous mixing is still not well understood, and need further study.
 In section 2, we found that Δre,3.7–2.1 is generally smaller (within ±2 μm) or even slightly positive over costal stratocumulus cloud regions (see Figure 9 and Figure 10). Clouds in these regions are relatively spatially homogeneous and usually non-precipitating. In situ measurements indicate that re generally increases from cloud base toward cloud top in low-level stratocumulus clouds [Martin et al., 1994; Miles et al., 2000]. The relatively small Δre,3.7–2.1 over these cloud regions could be the result of a combination of reasons, including adiabatic cloud structure, retrieval uncertainties and possibly cloud top entrainment process. However, none of these could explain the large negative Δre,3.7–2.1 over the broken cloud regions (see Figure 9 and Figure 10) and the strong dependence of Δre,3.7–2.1 on re,2.1 (Figure 11) and Hσ (Figure 13).
 Recently, the potential implications of the collision-coalescence and sedimentation processes on MODIS cloud property retrievals have been receiving increasing attention. Similar to the present study, Nakajima et al. [2010b, 2010a] also found that the MODIS re,3.7 retrieval for marine water clouds is substantially smaller than re,2.1 when re,2.1 is larger than about 15 μm. From collocated CloudSat observations, they noted that clouds with MODIS re,2.1 > 15 μm are often drizzling. Based on these observations, Nakajima et al. proposed that the cloud vertical structure caused by the drizzling process is responsible for the large difference between the MODIS re,2.1 and re,3.7 when re,2.1 larger than 15 μm. They further suggested that the increasing difference between re,2.1 and re,3.7 with increasing re (i.e., Figure 11f) is a result of increasing drizzle probability with increasing cloud droplet size. These studies have shed important light on the potential impacts of the collision-coalescence process on the vertical structure of marine water clouds and the consequent implications for MODIS retrieval. These studies have also shown the opportunities offered by the three independent MODIS re for cloud physics studies.
 In section 2, we noted that MODIS re,3.7 retrieval is substantially smaller than re,2.1 over those regions where open-cell clouds are prevalent (see Figure 9 and Figure 10). Both in situ measurements and CloudSat remote sensing data suggest that open-cell clouds are often drizzling [vanZanten and Stevens, 2005; Wood et al., 2008; Kubar et al., 2009]. Actually, some recent studies have suggested that precipitation (i.e., drizzle) plays a potentially important role in regulating the horizontal structure of marine water clouds [Wang and Feingold, 2009; Feingold et al., 2010]. All these studies, including ours, suggest the possibility that drizzle may have a role in causing the large difference between re,3.7 and re,2.1. However, it should be noted that drizzle could affect MODIS re retrieval in different ways, for example, by directly changing cloud verticals structure as suggested by Nakajima et al. [2010b, 2010a], or through a more indirect path by changing cloud morphology (i.e., horizontal heterogeneity) which in turn leads to 3-D radiative effect in MODIS retrieval as will be discussed in the following sections.
3.2. Horizontal Cloud Heterogeneity and 3-D Radiative Effects
 As shown in Figure 13, MODIS re,1.6 and re,2.1 retrievals for marine water clouds exhibit a sharp increase when sub-pixel inhomogeneity index Hσ exceeds a certain threshold, while re,3.7 shows no obvious correlation with Hσ. Since in-cloud path length distributions in the 3.7 μm band can be substantially shorter than for the shorter wavelength bands because of magnitude order greater droplet absorption [Platnick, 2001], this result seems to implicate a role for cloud horizontal heterogeneity and 3-D effect in causing re retrieval differences. The impacts of sub-pixel inhomogeneity on the retrieval of cloud re using the Nakajima-King method have been discussed in several previous studies [e.g., Kato et al., 2006; Marshak et al., 2006; Boeke, 2009; Wolters et al., 2010].
 A simple example to illustrate such an impact using 1-D radiative transfer is given in Figure 14. Here we assume that the fraction (f) of an otherwise plane-parallel MODIS pixel overlying a black surface is covered by cloud with τ1 = 2.87 and re = 16 μm, and that the rest of the pixel (i.e., 1 − f) is covered by cloud with τ2 = 30.76 and re = 16 μm. Also, we assume that the cloud thickness is much smaller than the pixel size so that horizontal transport is not significant and 1-D radiative transfer is applicable to both portions of the pixel. Ideally, if the reflectance look-up table (LUT) used for the retrieval is perfectly orthogonal, the retrieval would simply reduce to a linear interpolation problem, and one could expect the retrieved τ and re to be (τ1 + τ2)/2 and 16 μm, respectively. However, as shown in Figures 14a–14c, retrieval LUTs are not perfectly orthogonal, especially for small optical thicknesses. As a result, sub-pixel inhomogeneity can have a significant and non-intuitive impact on the re retrieval. For example, Figure 14a shows the situation when f = 0.5 and the 1.6 μm band is used for the re retrieval. In the figure, the thick black line indicates the re = 16 μm contour line in the LUT and the two asterisks indicate the locations of τ1 = 2.87 and τ2 = 30.76. The red diamond indicates the observation when f = 0.5. Clearly, the observation is below the re = 16 μm contour line, indicating a potential re retrieval larger than 16 μm. The underlying physics is such that the cloud reflectivity in the 1.6 μm band (R1.6 μm) is sensitive not only to re, but also to τ (i.e., ∂R1.6 μm/∂τ > 0), resulting in the concave shape of the constant re contour line. This in turn leads to an overestimation of the re retrieval when there is significant sub-pixel inhomogeneity. For the re,3.7 LUT, a thermal signal was added to the reflectance signal, with cloud and surface temperatures fixed at 275K and 290K, respectively; this explains the ordinate of Figure 14c being in units of radiance. The important point here is that the same level of sub-pixel inhomogeneity can affect the re,1.6, re,2.1 and re,3.7 retrievals to different extents and even in different directions. This point is further illustrated in Figure 14d, in which we retrieved the re,1.6 (blue line), re,2.1 (green line) and re,3.7 (red line) for f values varying from 0 to 1; the re,3.7 retrieval is made using a thermal correction identical to the MODIS C5 algorithm. We also calculated the sub-pixel inhomogeneity index Hσ, defined in equation (1), corresponding to each f value (gray dashed line). As expected, at both ends of Figure 14d, where the MODIS pixel is homogeneously covered by either τ1 = 2.87 or τ2 = 30.76, the three re retrievals are all close to the true value of 16 μm, except when f = 1 (i.e., pixel is fully covered by τ1 = 2.87), where re,3.7 is slightly larger due to the error caused by thermal correction. What is interesting, although not unexpected (see Figures 14a–14c), is that both re,1.6 and re,2.1 become larger than 16 μm as the pixel becomes inhomogeneous (i.e., 0 < f < 1), while the re,3.7 retrieval remains close to 16 μm for all f values. Moreover, it can be clearly seen that the difference between the three re retrievals generally increases with increasing sub-pixel inhomogeneity. This is because the three LUTs have different degrees of linearity and are therefore affected by the sub-pixel inhomogeneity to different extents. Overall, the simple experiment shown in Figure 14d clearly shows that the different sensitivities of the three re retrievals to the sub-pixel inhomogeneity could lead to substantial difference among them, and this difference increases with increasing sub-pixel inhomogeneity. These results are in agreement with the observations shown in Figure 13, as well as previous studies [e.g., Kato et al., 2006; Marshak et al., 2006; Boeke, 2009; Wolters et al., 2010].
 In the above example, the retrievals are based on conventional 1-D radiative transfer simulation. However, it is well known that 3-D radiative effects can have significant impacts on cloud retrievals [King et al., 1998; Davis and Marshak, 2010]. In order to test whether conclusions from the Figure 14 example still hold in a more realistic scenario, we take the experiment one step further by accommodating for 3-D effects in the forward radiative transfer simulations. The 3-D experiment is shown in Figure 15. In this case, we considered an idealized “step-function” cloud field, with periodic boundary conditions. The domain size is 16 km, which consists of 64 equal-width (250m) pixels along the x-direction. The cloud and surface properties are specified similar to the previous example. The center of the domain, from 4 km to 12 km, is covered by a cloud with τ = 30.76 and re = 16 μm, and the remainder of the domain (i.e., 0∼4 km and 8∼12 km) is covered by a thinner cloud with τ = 2.87 and re = 16 μm. The physical thickness of the cloud is assumed to be a constant of 0.5 km throughout the domain (i.e., flat cloud top and bottom). The cloud and surface temperatures are set at 275K and 290K, respectively. The surface is assumed to be black and atmospheric absorptions are ignored in all simulations for simplicity. The Sun is illuminating the domain from the left side to the right (+x direction) with a zenith angle of 50 degree. The 3-D radiative transfer simulations for this case were carried out using the community Monte-Carlo code developed by Pincus and Evans  as part of the I3RC (Intercomparison of 3-D Radiation Codes) project [Cahalan et al., 2005]. In the forward simulation, up-welling radiances at nadir were first computed at the center of every 250m pixel for the 0.86, 1.6, 2.1, 3.7 and 11 μm bands. Then, the 250m resolution radiances were averaged to 1 km resolution through an un-weighted moving average process. Figure 15a shows the averaged radiation at nadir. Several features can be observed. First, as expected the radiance fields are flat at the center of the cloud where the cloud is homogenous and the radiative transfer reduces to the conventional 1-D problem. Interestingly, the cloud reflectances, R(0.86 μm), R(1.64 μm) and R(2.1 μm), show a clear enhanced peak at the region from about 4∼5 km. This is the 3-D radiative effect, resulting from horizontal photon transportation [Várnai and Davies, 1999]. The same mechanism also leads to the shallow reduction of cloud reflectivity around 12 km. For a better comparison with the thermal emission part ((Iems (3.7 μm)), we converted the 3.7 μm cloud reflectivity to radiance Iref (3.7 μm) using the solar spectral irradiance from Platnick and Fontenla . The abovementioned 3-D radiative effects are still seen, although weaker, in the Iref (3.7 μm) field. For this particular case, because the cloud is relatively warm and cloud droplet size is large, the 3.7 μm band observation is dominated by the thermal emission part Iems (3.7 μm). This part is removed through a thermal correction process carried out in a manner consistent with the C5 MODIS algorithm. Based on the 3-D radiative transfer simulations shown in Figure 15a, we retrieved the re1.6, re,2.1 and re,3.7 using the operational 1-D plane-parallel LUTs. The retrieval results are shown in Figure 15b. Also plotted in Figure 15b is the sub-pixel inhomogeneity Hσ (gray dashed line) computed from 250m R(0.86 μm) fields. As expected, at the center of thick part of the cloud (e.g., 8 km), the three re are all very close to the 16 μm re assumed in the forward simulation. At the center of the thin part of the cloud (e.g., 14∼16 km), re,3.7 is slightly larger than re,1.6 and re2.1. Similar to the 1-D case in Figure 14, this overestimation of re,3.7 is due to the error caused by the thermal correction process. At about 4∼5 km, due to the aforementioned horizontal photon transport, the cloud “appears” brighter in the 1.6 μm and 2.1 μm bands than it would appear in the plane-parallel condition (see Figure 15a). This explains the underestimation of re,1.6 and re,2.1 in this region. What is particularly interesting is that both re,1.6 and re,2.1 show pronounced peaks over the transition regions (e.g., 3∼4 km and 11∼12 km), as they also did in the 1-D case in Figure 14. The behavior of re,3.7 over these transition regions is also understandable. From 11 to 12 km, the R(0.86 μm) reduces quickly from about 0.6 to about 0.1, while the I(3.7 μm), including both Iref (3.7 μm) and Iems (3.7 μm), does not vary much. Looking again at the 3.7 μm LUT in Figure 14c, we find that such variation of R(0.86 μm) and I(3.7 μm) would result in an increased re,3.7 retrieval. From 3 km to 4 km, R(0.86 μm) increased quickly from about 0.1 to about 0.8, while the I(3.7 μm) decreased from about 0.35 to 0.3. The shape of the 3.7 μm LUT in the Figure 14c indicates that such a change would lead to a decreased re,3.7 retrieval as seen in see Figure 15b. Finally, it is worthy of special note that in the transition regions the differences among the three re retrievals, especially the difference between re,2.1 (re,1.6 as well) and re,3.7 generally increases with the increasing sub-pixel inhomogeneity. This aligns well with the results from 1-D case in Figure 14, as well as the observations in Figure 13.
 The above two cases show that the re,3.7 retrieval is less affected by 3-D effects in comparison with the other retrievals. This is because the stronger water absorption in the 3.7 μm band limits the horizontal transport of radiation and also makes the LUT of 3.7 μm band more orthogonal, which makes the 3.7 μm band retrieval less susceptible to the 3-D effects. Because of the different sensitivity to 3-D effect, the difference between re,3.7 and re,2.1 (re,1.6 as well) increases with increasing cloud heterogeneity. These results, along with the observational evidence (i.e., Figure 13), seem to indicate the potential role of cloud heterogeneity and 3-D radiative effects in causing the difference between re,2.1 and re,3.7 retrievals for marine water clouds. Admittedly, the above two cases are far from realistic as they are intended to provide theoretical perspective. Future work based on more realistic cloud models from, for example, large eddy simulations, will be helpful for improving our quantitative understanding of this issue.
3.3. Discussion of Possible Coupling of Cloud Vertical Structure and 3-D Effects
 As discussed above, both cloud vertical structure and 3-D effect can lead to significant differences between re,2.1 and re,3.7 for marine water clouds. In this section, we discuss the possibility that these two mechanisms are coupled and work together to cause large Δre,3.7–2.1 in some highly inhomogeneous clouds. Note that the two mechanisms have very different implications for the interpretation of MODIS observations. Therefore, it is useful to identify regimes where one mechanism plays a more dominant role than the other. However, it is also extremely difficult to separate the two mechanisms using MODIS observations alone since one, or both, of the mechanisms could affect the observations and their occurrence may in fact be correlated. Nevertheless, an attempt has been made to do just this. Figure 16 shows a color contour of the mean value of Δre,3.7–2.1 projected on sub-pixel heterogeneity index (Hσ) and re,2.1 for water clouds with τ > 5. The gray contour lines indicate the joint frequency histogram of Hσ and re,2.1 for one month of MODIS observation. Several interesting features are noteworthy in Figure 15. First of all, re,3.7 is almost always smaller than re,2.1. Second, the magnitude of the Δre,3.7–2.1 clearly increases with increasing re,2.1 over the entire Hσ space. Third, the largest Δre,3.7–2.1 (more than 12 μm) is found where both Hσ and re,2.1 are large. These features in Figure 16 give us some hints about the relative importance between cloud vertical structure and 3-D effect in casing Δre,3.7–2.1 for different types of clouds. As the contours indicate, the majority of marine water clouds with τ > 5 have Hσ smaller than 0.1. The 3-D radiative effects for these rather homogenous clouds can be expected to be less significant than for those highly inhomogeneous clouds. This seems to imply that the cloud vertical structure play a more important role in causing the difference between re,3.7 and re,2.1 in these clouds. Figure 16 also indicates that a non-negligible fraction of marine water clouds are highly inhomogeneous and have large effective radius according to MODIS. For these clouds, MODIS retrieves very different re,3.7 and re,2.1.
 It would not be surprising to find that the occurrence of drizzle and heterogeneous cloud fields are correlated in many cases. For example, Wang and Feingold  and Feingold et al.  found that drizzle plays an important role in regulating the horizontal structure of marine water clouds. An implication of such findings is that even if the direct radiative effect of drizzle drops is small, drizzle processes may still indirectly cause large Δre,3.7–2.1 by modifying cloud horizontal structure, which in turn leads to 3-D radiative effects. Figure 16 indicates that interpretation of Δre,3.7–2.1 for marine water clouds is a subtle issue, as it could be a result of multiple coupled mechanisms. Finally, it is worth mentioning that the result remains the same if we include thin clouds (τ < 5) in Figure 16.
4. Conclusions and Summary
 The MODIS instrument provides three different retrievals of cloud effective radii based on the 1.6 μm, 2.1 μm and 3.7 μm channels. Understanding the differences found among these retrievals for marine water clouds is particularly important, because retrieval spectral discrepancies may provide useful information for cloud process studies. Based on one month of Terra MODIS data (April 2005), we performed a systematic assessment of the difference among re,1.6, re,2.1 and re,3.7 for marine liquid water clouds. The main findings from this assessment are summarized as follows:
 1. Significant differences are often found between re,2.1 and re,3.7, and the differences are a strong function of cloud regime. The two retrievals are in reasonable agreement over coastal stratocumulus regions, but differ substantially over the broken trade wind cumulus cloud regions, with re,3.7 biased low (Figure 9 and Figure 10).
 2. Correlation studies reveal that the difference between re,2.1 and re,3.7 are relatively small for clouds with re,2.1 smaller than about 15 μm, but the difference increases quickly after re,2.1 exceeds 15 μm (Figure 11).
 3. The differences between the three re retrievals for thin clouds are large and random, indicating large retrieval uncertainties (Figure 12).
 4. The correlation studies also reveal that both re,1.6 and re,2.1 have a clear dependence on cloud heterogeneity index Hσ, calculated from high resolution 0.86 μm band reflectance observation (Figure 13). When Hσ is small, the PDF of both re,1.6 and re,2.1 remain relatively stable. When Hσ exceeds a certain threshold (∼0.3), both re,1.6 and re,2.1 are found to increase with Hσ . In contrast, the re,3.7 retrieval shows no such threshold-like behavior. The different sensitivities of re,3.7 and re,2.1 to Hσ aligns with the increasing difference between the two with increasing Hσ. The mechanisms through which cloud horizontal heterogeneity and 3-D radiative effects could cause the difference between the three re retrievals were discussed using a based on both 1-D (Figure 14) and 3-D (Figure 15) radiative transfer simulations.
 These findings point to the fact that cloud effective radius retrievals from passive satellite-sensors are dependent on the choice of spectral band used in the algorithm. An important implication of this dependence is that the re retrievals based on different spectral bands seem not to be directly comparable. For example, comparing MODIS Level-3 gridded liquid water cloud re (based on the 2.1 μm retrieval) with algorithms that use a 3.7 μm band (e.g., AVHRR PATMOS-x, Heidinger et al.; MODIS-CERES team [Minnis et al., 1998]) should be undertaken with caution. We also speculate that the difference between re retrievals could be a combined result of cloud vertical structure and 3-D radiative effects. It should be noted that the objective of this study is not to argue which re retrieval is better, but to understand how re retrievals based on different spectral band are affected by cloud 3-D structure and 3-D radiative transfer process. Each re retrieval from MODIS has its own advantages and limitation. For example, the re,3.7 is less sensitive to the 3-D radiative effect, but at the same time it is subject to uncertainties associated with the thermal correction. Although re,1.6 is significantly affected by cloud heterogeneity and 3-D effect, it penetrates deeper into cloud and therefore could be useful for detecting drizzle. The combination of these re retrievals, together with other complementary observations like CloudSat precipitation product, may help understand some critical issues in cloud and climate studies, such as drizzling process and aerosol indirect effect [e.g., Nakajima et al., 2010b, 2010a; Suzuki et al., 2010]. However, results from this study also indicate that we have to go beyond the simple plane-parallel homogenous cloud model to fully understand the potential and limitations of MODIS re retrievals. One direction for future research to simulate MODIS cloud property retrievals from high-resolution cloud-resolving models or large-eddy simulation models. The comparison between the synthetic retrievals and the original cloud fields from cloud-resolving models can help reveal the information content and limitations of MODIS cloud property retrievals, under realistic cloud conditions.
 We would like to thank Brent Maddux and Steve Ackerman for many helpful discussions. We also thank Gala Wind for providing research-level MOD06 code and Kerry Meyer for proofreading the manuscript and providing helpful suggestions. Z.Z. acknowledges NASA funding support under the grant NNX11AI98G. Z.Z. would like to thank Alexander Marshak and Tamas Varnai for valuable discussion on 3-D radiative effect, and Robert Pincus for his help on I3RC code. This work was funded in part by NASA’s Radiation Sciences Program.