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].
Figure 1. True-color image (R: band 1; G: band 4; B: band 3) of a data granule collected by Terra MODIS on April 2nd, 2005 over the South-Eastern Pacific. Note that clouds in this granule showed a clear east-to-west transition from closed-cell stratocumulus cloud to open-cell cumulus clouds. Courtesy MODIS Atmosphere Team (modis-atmos.gsfc.nasa.gov/IMAGES/).
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 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].
Figure 2. (a) The cloud topical thickness and (b) effective radius retrievals for the data granule shown in Figure 1. The maps of (c) Δre,1.6–2.1 and (d) Δre,3.7–2.1. (e) The probability density functions of the re,1.6, re,2.1 and re,3.7. (f) The PDFs of Δre,1.6–2.1 and Δre,3.7–2.1.
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 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 3. (a) the cloud horizontal heterogeneity index (Hσ) defined in equation (1) for the granule shown in Figure 1. (b) The PDFs of Δre,1.6–2.1 in different intervals of Hσ. (c) Same as Figure 3b but for Δre,3.7–2.1.
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 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).
Figure 4. True-color image of Terra MODIS data granule on April 2nd, 2005 over the Central Pacific. The center area of this granule is covered mostly by broken cumulus clouds. The northeast and southwest corners are covered by ice clouds. Courtesy MODIS Atmosphere Team (modis-atmos.gsfc.nasa.gov/IMAGES/).
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 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.
Figure 7. (a) Classification of the pixels in Figure 4 when the Clear-Sky Restoral (CSR) algorithm is turned on in the MODIS retrieval. (b) Classification when CSR is turned off. The behavior of re,1.6 for the CSR pixels is very similar to that of re,2.1 in this figure, and therefore not shown here. Note that the re,1.6, re,2.1, and re,3.7 retrieval counts increased in different amounts when CSR is turned off. The successful retrieval counts are summarized in Table 1.
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Figure 8. (top) The PDFs of re,1.6 (blue), re,2.1 (black), and re,3.7 (red) for the water clouds pixels in Figure 4, sampled by the condition that all three re retrievals must be successful when the Clear-Sky Restoral (CSR) algorithm is on. (middle) The PDFs when the retrievals are sampled independently and CSR is on. (bottom) The PDFs when retrievals are sampled independently and CSR is turned off; the dashed lines correspond to additional water pixels gained by turning CSR off.
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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|
 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.