Regional Biases in MODIS Marine Liquid Water Cloud Drop Effective Radius Deduced Through Fusion With MISR

Abstract Satellite measurements from Terra's Moderate Resolution Imaging Spectroradiometer (MODIS) represent our longest, single‐platform, global record of the effective radius (Re) of the cloud drop size distribution. Quantifying its error characteristics has been challenging because systematic errors in retrieved Re covary with the structural characteristics of the cloud and the Sun‐view geometry. Recently, it has been shown that the bias in MODIS Re can be estimated by fusing MODIS data with data from Terra's Multi‐angle Imaging SpectroRadiometer (MISR). Here, we relate the bias to the observed underlying conditions to derive regional‐scale, bias‐corrected, monthly‐mean Re 1.6, Re 2.1, and Re 3.7 values retrieved from the 1.6, 2.1, and 3.7 μm MODIS spectral channels. Our results reveal that monthly‐mean bias in Re 2.1 exhibits large regional dependency, ranging from at least ~1 to 10 μm (15 to 60%) varying with scene heterogeneity, optical depth, and solar zenith angle. Regional bias‐corrected monthly‐mean Re 2.1 ranges from 4 to 17 μm, compared to 10 to 25 μm for uncorrected Re 2.1, with estimated uncertainties of 0.1 to 1.8 μm. The bias‐corrected monthly‐mean Re 3.7 and Re 2.1 show difference of approximately +0.6 μm in the coastal marine stratocumulus regions and down to approximately −2 μm in the cumuliform cloud regions, compared to uncorrected values of about −1 to −6 μm, respectively. Bias‐corrected Re values compare favorably to other independent data sources, including field observations, global model simulations, and satellite retrievals that do not use retrieval techniques similar to MODIS. This work changes the interpretation of global Re distributions from MODIS Re products and may further impact studies, which use the original MODIS Re products to study, for example, aerosol‐cloud interactions and cloud microphysical parameterization.


Introduction
The effective radius (Re) of the cloud drop size distribution plays an important role in the energy and water cycle of the Earth system (Platnick & Twomey, 1994;Twomey, 1991) and is listed as an Essential Climate Variable by the Global Climate Observing System (GCOS, 2011). Satellite-retrieved Re has extensively been used, for example, to assess aerosol-cloud interactions (e.g., Menon et al., 2008) and to evaluate cloud parameterization in climate models (e.g., Ban-Weiss et al., 2014;Suzuki et al., 2013). Owing to its importance, stringent requirements on Re accuracy of better than 10% have been called for (e.g., Ohring et al., 2005). However, comparison between currently available satellite-derived Re products (e.g., Bréon & Doutriaux-Boucher, 2005;Sayer et al., 2011;Stubenrauch et al., 2012) reveal disparities ranging from~2 μm (~20%) to~9 μm (~50%) in regional monthly-mean values.
The Re products from the Moderate Resolution Imaging Spectroradiometer (MODIS) are widely used owing to its global coverage and long record-now over 18 years. MODIS uses two shortwave spectral channels to retrieve Re and cloud optical depth simultaneously (Platnick et al., 2003). This bispectral retrieval technique makes several assumptions, including 1-D radiative transfer, single-mode drop size distribution, and cloud horizontal and vertical homogeneity. Deviations from these assumptions in nature lead to potential bias in the retrieved Re. For example, Marshak et al. (2006) showed that the deviations from the 1-D radiative transfer assumption can lead to overestimates of retrieved Re of up to a factor of 2 for simulated cumulus clouds, while Werner et al. (2016) showed case studies in which the difference between the MODIS Re and that deduced using a coincident higher-resolution imager can exceed 10 μm for partially cloudy pixels-but with significant scatter having both positive and negative biases.
Comparison between MODIS-retrieved Re and in situ observations shows that overestimates in MODIS Re range from about −0.22 to 13 μm, depending on cloud type, Sun-view geometry, the choice of in situ probe, and the details of the cloud microphysical properties (e.g., Ahn et al., 2018;Glienke et al., 2017;King et al., 2013;Painemal & Zuidema, 2011;Witte et al., 2018). In particular, Witte et al. (2018) suggested that while satellite retrievals are commonly validated against in situ measurements, the uncertainty of the aircraft retrievals should be acknowledged. They showed that in situ measurements of the MODIS Re bias may be overestimated. Nevertheless, these and many other studies presented tremendous insight into the issues in retrieving Re in the face of cloud heterogeneity, as well as in the quality of MODIS-retrieved Re under certain conditions, none characterized the MODIS Re bias at regional scales over the entire globe. Recently, progress was made in Liang et al. (2015), where they characterized the latitudinal bias in MODIS-retrieved Re through data fusion of the Multi-angle Imaging Spectroradiometer (MISR; Diner et al., 1998) and MODIS, both onboard the Terra satellite. They showed that the zonal-mean bias (i.e., averaged across longitude at a given latitude) in MODISretrieved Re for marine liquid clouds varied from 2 to 11 μm, with smaller biases occurring at latitudes where stratiform clouds contribute the most to total cloud fraction.
In this study, we extend the approach of Liang et al. (2015) to estimate the monthly-mean bias in the MODISretrieved Re at regional scales. We develop a bias-correction method that relates the bias to observed cloud properties and Sun-view geometry and apply it to 8 years (2001)(2002)(2003)(2004)(2005)(2006)(2007)(2008) of January and July MODIS-retrieved Re from the 1.6, 2.1, and 3.7 μm spectral channels, respectively, namely, Re 1.6 , Re 2.1 , and Re 3.7 . We show that the differences among these three bias-corrected Re products are greatly reduced compared to the original products and that the remaining differences may indicate vertical variations of Re that appears to be cloud-type dependent. Comparisons between bias-corrected Re and those derived from other satellite instruments, field observations, and climate models are also discussed.

Data and Method
Fusion of the Terra MISR and MODIS data at cloud top is described in detail in Liang et al. (2009) and Liang and Di Girolamo (2013). We use the MODIS Collection 6 Cloud Product (MOD06; Platnick et al., 2015) and Version 24 of the MISR L1B2 ellipsoid-projected radiance product from 2001 to 2008 for the months of January and July in order to provide ample sampling for good statistical characterization and to contrast two opposing seasons. Only liquid water clouds were considered based on the cloud phase flag in MOD06. Cloud optical depth (τ) at the MISR view geometries are retrieved using MISR near-infrared BRF and MODIS-retrieved Re as described in Liang et al. (2009). The fusion between MISR and MODIS data requires projecting MODIS Re and τ retrievals onto the 1.1 km MISR SOM (Space Oblique Mercator) grid. To reduce registration errors of clouds across MISR cameras, pixels are grouped in 3 by 3 1.1 km pixel domains, followed by the M2 feature matcher (Muller et al., 2002) to track clouds across MISR's nine cameras. The quality control procedure requires that a given 3 by 3 pixel domain has successful coregistration across MISR's nine camera views and that valid MODIS Re and τ retrievals are identified as fully cloudy for the 3 by 3 domain (see Liang et al., 2009 for details on the quality control procedures, with further discussion in Liang & Di Girolamo, 2013). Following the quality control procedures, a total of 48.2% and 51.4% of all fully cloudy, valid MODIS retrievals are used in this study for January and July, respectively. The distribution of these MODIS Re values compared to the total population of MODIS Re values was shifted about 1 μm larger, but with no discernible differences in the spatial pattern. The distribution of τ for these data shows no shift in the mode of the distribution, but rather a narrowing of the distribution relative to the total population of MODIS τ values, particularly for large τ values that are often found at high latitudes in the winter hemisphere.
Our estimate of the Re bias in the MODIS products begins with the approach described in Liang et al. (2015). In brief, Liang et al. (2015) showed that cloud optical depth retrievals as a function of observed scattering angle exhibit a local minimum in the vicinity of the cloud-bow scattering angle (~140°) whenever the Re is overestimated. They revealed that these local minima, referred to as τ-rainbow dips (although more appropriately τ-cloudbow dips), are prevalent within the MODIS cloud optical depth product, as well as in the MISR cloud optical depth retrievals that use MODIS Re as an input. They demonstrated that the amplitude of the τ-rainbow dip can be used to estimate the magnitude of the Re bias in the MODIS products. However, depending on which side of the cloud bow is used to measure the amplitude of the τ-rainbow dip (e.g., 135-140°vs. 140-145°), the bias is either overestimated or underestimated because of broader variations in the view-angle dependence of retrieved cloud optical depth (e.g., Liang & Di Girolamo, 2013). Owing to MISR's unique ability to sample scattering angles on either side of the rainbow at many latitudes, they were able to produce upper and lower bound estimates of the MODIS Re bias at those latitudes, showing latitudinal variations in zonal-mean Re that were consistent with expected biases caused by the departure from the homogeneous cloud assumption for clouds typically found at the observed latitudes (e.g., latitudes dominated by stratiform vs. cumuliform clouds). A separate study examined the impact of the vertical variation in Re and drizzle on the method of Liang et al. (2015) for estimating the bias in MODIS-retrieved Re, showing that these two factors contribute no more than~1 μm to the bias estimates over a wide range of drizzle and vertical variations in Re (Su, 2017). The potential impact of a fixed cloud droplet size distribution on the Re retrievals is also another source of error. As shown in previous studies (e.g., Benas et al., 2019;Bréon & Doutriaux-Boucher, 2005;Chang & Li, 2002), the error in retrieving Re from the bispectral technique, such as used by MODIS, due to deviations from the assumed effective variance of the droplet size distribution is typically less than~1 μm. The differences in the scattering angular variation in retrieved Re across the cloud bow (important to the method of Liang et al., 2015) between a range of effective variance are even smaller. This error does propagate to our results and, as shown below, is small relative to the difference between upper and lower bound estimates of the MODIS Re bias.
Here, we relate the upper and lower bound estimates of the MODIS Re bias had by the method of Liang et al. (2015) to other variables that can be measured by MODIS alone. This provides the basis for constructing a MODIS Re bias-correction procedure, which we then apply to MODIS data to obtain regional estimates of bias-corrected Re of marine liquid water clouds. The choice of variables is based upon our current understanding of potential factors that may contribute to the Re bias due to three dimensional (3-D) radiative effects: solar-zenith angle (SZA), cloud heterogeneity, and cloud optical depth (e.g., Grosvenor & Wood, 2014;Loeb & Davies, 1996;Marshak et al., 2006;Zhang et al., 2012). We did examine additional variables, such as cloud top height, cloud top temperature, and above-cloud water vapor, but they showed no significant relationship to the observed MODIS-retrieved Re bias (not shown).
The fused MISR-MODIS data are stratified into five MISR nadir τ bins (0-4, 4-12, 12-16, 16-24, and >24) and seven cloud heterogeneity-metric (H σ ) bins (0-0.02, 0.02-0.04, 0.04-0.06, 0.06-0.08, 0.08-0.1, 0.1-0.2, and >0.2). These bin widths were chosen to loosely provide similar number of samples in each bin and to ensure good separation between the optically thin and smooth clouds (characterized by small τ and H σ ) from the optically thick and rough clouds (characterized by large τ and H σ ). Starting with MODIS Collection 6 , H σ is defined as the standard deviation divided by the mean 0.87 μm reflectance of 4 × 4 250 m resolution pixels within a 1 km MODIS footprint (Liang et al., 2009). As in Liang et al. (2015), the data are further grouped into 2.5°latitude bins to capture latitudinal variations and 1°SZA bins within each latitude bin. The method of Liang et al. (2015) retrieves a Fc correction factor, such that Re corrected = Fc × Re MODIS , where Re corrected is the "bias-corrected" Re value and Re MODIS is the original MODIS-retrieved Re value. This is done for MODIS-retrieved Re from the 1.6, 2.1, and 3.7 μm spectral channels, thus a separate Fc value is produced for each channel. For each latitude bin, Fc across all available SZA bins are retrieved, from which the mean, Fc, and standard deviation, σ Fc , are calculated for the upper and lower bounds of estimated Fc. The reported Fc and σ Fc therefore are both dependent on τ, H σ , latitude, month, and MODIS channel.
Along with stratifying the data by latitude, we also experimented with a bias-correction approach using SZA, rather than latitude, as a dependent variable along with curve fitting rather than binning for the correction factors. The two approaches showed similar global distribution of bias-corrected monthly-mean Re values (section 3 below), with an overall global RMS difference of 0.6 μm and essentially unbiased. We present the latitude-binning approach because (1) it is more tightly tied to the region, whereas the same SZA in a given month for Terra samples can occur both north and south of the solar equator, and (2) it provides σ Fc used for error propagation. Further characterization by cloud regime is naturally captured by using H σ and τ as dependent variables for the bias correction. We also examined the difference of stratifying by using both H σ and τ against using only H σ or τ. Our results (not shown) indicate that the inclusion of H σ in the stratification process captures significantly more of the variability across the globe than stratification by τ alone.
Since some latitudes do not contain the required MISR-observed scattering angles around the rainbow, Fc is linearly interpolated between latitude bins with valid retrievals (Liang et al., 2015); these latitude bins are identified in the figures in section 3. For January, camera pairs were restricted to only adjacent neighboring MISR camera pairs (i.e., An-A, A-B, B-C, and C-D camera pairs for forward and aft directions). In July, far fewer latitude bins had adjacent MISR cameras observing scattering angles on both sides of the rainbow dip, so non-adjacent camera pairs were also used for Fc retrievals at the expense of producing more widely separated upper and lower bound estimates of the MODIS Re bias (Liang et al., 2015).
Following the data stratification described above, we retrieve Fc for each τ-H σ bin to provide bias correction for each 2.5°latitude and longitude grid: where Re correct (lat,lon) is the corrected mean Re value and Re MODIS(lat,lon) is the mean MODIS Re value for each 2.5°grid, Fc lat ð Þ is the upper or lower bound mean Fc for each 2.5°latitude bin, and i is the index for each τ-H σ bin (i = 1, 2, … , 35). For a given month, the final bias-corrected monthly-mean Re for each 2.5°grid point is weighted by the number of Re samples: where Re w_correct (lat,lon) is the weighted-mean corrected Re for each 2.5°grid and N (lat,lon) (i) is the number of Re samples that passed the quality control for each 2.5°grid for bin i.
The standard error in Re w_correct (lat,lon) can be shown to equal: where σ SEcorrect lat;lon ð Þ is the standard error of Re w_correct (lat,lon) , σ (lat) (i) is the standard deviation of upper or lower bound Fc at 2.5°latitude resolution, and n (lat) (i) is the number of Fc samples at 2.5°latitude resolution for bin i.

Bias-Corrected MODIS Re
Figure 1 shows our examination of Re 2.1 for the month of January averaged between 2000 and 2008. Figure 1 includes the monthly mean of MODIS Re 2.1 , Re w_correct2.1 using the upper bound of Fc (minimum estimate of the bias) and lower bound of Fc (maximum estimate of the bias), the difference between upper and lower bound estimates, the minimum MODIS Re 2.1 -bias based on the upper bound of Fc, and H σ -all sampled within the MISR nadir-camera swath. Results for Re 1.6 and Re 3.7 are provided in Figures 2 and 3, respectively. Figures 1a, 1c, and 1e reveal large differences between the MODIS Re 2.1 and Re w_correct2.1 : The range of monthly mean Re values for MODIS Re 2.1 is~10 to 25 μm, while it is~4 to 13 and~5 to 17 μm for the lower and upper bounds of Re w_correct2.1 , respectively. These produce area-weighted, global (60°N to 60°S; oceanonly) mean values of 16.6, 11.2, and 7.9 μm for MODIS Re 2.1 , upper bound Re w_correct2.1 , and lower bound Re w_correct2.1 , respectively. The standard error (Figures 1d) for the upper bound ranges from~0.1 to 1.8 μm, with larger errors primarily due to regions with lower sampling in Fc retrievals (latitude bands missing Fc retrievals are shown in white, indicating that those regions are processed with interpolated Fc values as discussed in section 2). Similar values are had for the standard errors for the lower bound Re w_correct2.1 (not shown). These error estimates are mostly much smaller than the differences between the upper and lower bound estimates of Re w_correct2.1 shown in Figure 1f. The area-weighted, global mean difference between upper and lower bound estimate of Re w_correct2.1 is 3.2 μm, with regional differences ranging from~1 to 6 μm. In the following discussion, we mainly focus on the bias associated with the upper bound Re w_correct2.1 estimates (which gives minimum estimates of the bias in Re 2.1 in Figure 1g)-the actual bias that the estimated bias in Re 2.1 has a strong dependence on cloud regime, ranging from~1 to 3 μm in regions with more homogeneous clouds (e.g., marine-stratocumulus regions off the west coasts of South America, Africa, and Australia) to~8 to 10 μm in regions with more heterogenous clouds (e.g., 5°S to 10°N in the western Pacific Ocean). This is consistent with our knowledge that greater 3-D radiative effects that cause, on average, larger biases in the retrieval of Re using the bispectral technique occur for more heterogeneous clouds (e.g., Marshak et al., 2006). This is also reflected when compared to H σ in Figure 1b, particularly in regions where differences in upper and lower bound estimates of Re w_correct2.1 are small (Figure 1f). In terms of relative bias (the ratio between the absolute bias and the MODIS Re 2.1 ), Figure 1h shows that regions with large absolute biases do not necessarily correspond to regions with large relative biases. We see large bias (~40 to 60%) in regions around 30°S to 45°S, 5°S to 10°N, and 40°N to 50°N, and still small bias (~15 to 20%) in the coastal marine-stratocumulus regions. Some zonal striping is evident in Figures 1c-1h. These may arise from several sources, including the smaller number of samples for some latitude bins, the zonal nature of solar zenith angles tied to Terra's 10:30 a.m. equator crossing time, and, to some degree, the zonal nature of the general circulation of the atmosphere. Similar to Figure 1, Figures 2 and 3 show the correction for Re 1.6 and Re 3.7 , respectively, for the month of January averaged between 2000 and 2008. Along with Figures 1a, Figures 2a and 3a reveal clear discrepancies between the original MODIS Re 1.6 , Re 2.1 , and Re 3.7 : the range of monthly mean Re values for MODIS Re 1.6 , Re 2.1 , and Re 3.7 is~9 to 22,~10 to 25, and~8 to 18 μm, respectively. After applying the bias correction procedures, much more consistent estimates among the three channels are revealed: with monthly mean Re w_correct values generally within~1 to 2 μm among the three channels. The area-weighted, global mean values for MODIS Re, upper bound Re w_correct , and lower bound Re w_correct for the three spectral channels are provided in Table 1. The difference between the upper and lower bound estimates of Re w_correct1.6 ( Figure 2f) and Re w_correct3.7 (Figure 3f) are comparable to that of Re w_correct2.1 (Figure 1f) both in terms of spatial patterns and magnitudes. Figure 2 shows that the upper bound estimates of Re 1.6 bias range from 1 to 9 μm (Figure 2g), corresponding to~15 to 60% in relative Re 1.6 bias ( Figure 2h). However, Figure 3 shows that the upper bound estimates of Re 3.7 bias ranges from~1 to 6 μm (Figure 3g), corresponding tõ 12 to 45% in relative Re 3.7 bias (Figure 3h), which is somewhat smaller than the upper bound estimates of Re 2.1 bias and Re 1.6 bias. This is consistent with the findings in Zhang and Platnick (2011) that Re 3.7 is less susceptible to 3-D radiative effects compared to the other two channel retrievals. Further discussion on spectral channel differences of retrieved and corrected MODIS Re are discussed in section 3.3. Figure 4 displays the same information as Figure 1, but for July. Note that more latitudes have missing Fc retrievals, as indicated in Figures 4d, compared to January owing to less favorable scattering angles sampled by MISR for our technique (section 2). Figure 4a reveals the July monthly mean Re values for MODIS Re 2.1 is 9 to 21 μm, while it is~3 to 14 and~6 to 17 μm for the lower and upper bounds of Re w_correct2.1 , respectively, after bias correction. The area-weighted, global mean values for July are 16.3, 12.1, and 7.6 μm for MODIS Re 2.1 , upper bound Re w_correct2.1 , and lower bound Re w_correct2.1 , respectively. As noted above, the standard error for the upper bound estimates of Re w_correct2.1 in July have more latitudes with missing Fc retrievals, and the magnitude of the standard error is consistent with that of the January estimates. The difference between the upper and lower bound estimates of Re w_correct2.1 for July, however, reveals significant differences when compared to the January results: From Figure 4f, we observe large upper and lower bound differences of up to~5 to 9 μm around 10°N to 15°S and 50 to 70°N, and much closer upper and lower bounds of 1 to 4 μm around 15 to 30°N and 25 to 60°S. The large upper-lower bound differences coincide with the latitudes that have missing Fc retrievals, which indicates that the lower sampling at the latitude boundaries of the missing Fc retrievals simply leads to larger differences between upper and lower bounds.
As in January, regional overestimates in the mean MODIS Re 2.1 of at least~1 to 10 μm (at the upper bound estimates of Re w_correct2.1 ) are observed for July (Figure 4g), but with different regional patterns owing to differences in the prevailing cloud heterogeneities (Figures 1b vs. 4b) and solar zenith angles. Specifically, the relative biases in Figures 1h and 4h reveal interesting patterns: Large relative bias along~10°N is consistent with convective clouds along the ITCZ, and the large biases in the tropics spreading southward in January and northward in July are in line with the seasonal shift in the strength of the Hadley cell (e.g., Dima & Wallace, 2003). This also correlates well with tropical H σ values in Figures 1b and 4b, since larger Re biases are expected to occur for more heterogeneous clouds. At middle to high latitudes in the Southern Hemisphere, January shows a band of large Re biases between~35°S and 40°S (large H σ in Figure 1b) and small Re biases within~45°S to 60°S (small H σ in Figure 1b), indicating that in moderately high SZAs, the

10.1029/2019JD031063
Journal of Geophysical Research: Atmospheres texture of clouds (quantified by H σ ) dominates the zonal variations in Re bias more than SZA effects. In July, however, low Re biases are observed from~30°S to 45°S, while bias increase substantially between 45°S and 60°S despite no corresponding increase in the moderately large H σ values, indicating the dominant effects of large SZAs (e.g., from shadows) on the gradient in the bias. Such rapid increases in retrieval errors with increasing SZAs larger than~70°for heterogeneous clouds have been discussed in other studies (e.g., Grosvenor & Wood, 2014;Loeb et al., 1997). One curious feature in Figure 4h is that the Re 2.1 bias for July is low in~30°S to 45°S (SZA~55°to 70°) relative to regions to the north and south. This may be due to competing 3-D-radiative pathways that produce a minimum in retrieval errors at midrange SZAs (e.g., Kato & Marshak, 2009;Várnai & Davies, 1999

Comparison to Other Data Sets
The Re biases reported in Figures 1g and 4g compare very well with the few existing in situ validation studies of MODIS Re 2.1 . For example, using data from the VAMOS Ocean-Cloud-Atmosphere-Land Study Regional Experiment (VOCALS-REx) in the Southeast Pacific during October to November 2008 (region in Figure 1g), Painemal and Zuidema (2011) showed an~1 to 2 μm bias in MODIS Re 2.1 ; from the Rain in Cumulus over the Ocean (RICO) field campaign from November 2004 to January 2005 (region in Figure 1g), Haney (2013) showed MODIS Re 2.1 biases ranging from~7 to 12μm, and aircraft observations in the southern ocean (region in Figure 4g) from June to October in 2013 to 2015, Ahn et al. (2018) showed an overestimate in MODIS Re 2.1 of~13 μm on average for nondrizzling clouds. All of these studies line up nicely with the regional Re 2.1 biases shown in Figures 1g and 4g. We also note that Miles et al. (2000) provided a review of field observations of marine liquid water clouds over the past 50 years indicating Re typically ranging from~4 to 15 μm, which is in line with Re w_correct2.1 . Similar conclusions are drawn in comparison with Re w_correct1.6 and Re w_correct3.7 .
We also compare the upper bound estimates of Re w_correct reported in Figures 1, 2 et al., 1994) from January 2006-2013. All data but POLDER were retrieved from the GEWEX Cloud Assessment Database (Stubenrauch et al., 2013). For POLDER, only Level 2 Re retrievals with a quality index >2.2 (Bréon & Doutriaux-Boucher, 2005) are averaged to a 2.5°resolution grid to ensure fair global coverage and sufficient sampling for analysis. Stubenrauch et al. (2012) has shown large differences in Re 2.1 among all these datasets with regional differences of around −7 to +9 μm in monthly means, but without exploring why they are different. Here, we focus on regions with POLDER sampling only since it is believed that POLDER provides superior retrievals owing to its advanced polarimetric approach, which is less sensitive to biases arising from 3-D effects (Bréon & Doutriaux-Boucher, 2005). All satellite retrievals are sampled to a 2.5°resolution grid and filtered to only include the grids that have sufficient POLDER coverage. Figure 5 reveals that, despite the independent nature of the various satellite instruments and the differences in sampling, a clear pattern emerges when organized by retrieval methods (bispectral methods on the right and other methods on the left): Panels in the left column share very similar regional patterns of Re, while panels on the right also show similar regional patterns but with much larger regional differences among each other. When compared with POLDER Re, the mean difference of monthly averages and RMS differences (in bracket) are provided in Table 2. The~3 μm difference of Re MODIS2.1 compared to POLDER Re is close to the~2 μm difference reported in Bréon and Doutriaux-Boucher (2005), where the~1 μm discrepancy may be due to different data collections (MODIS Collection 4 vs. 6), our sampling to within the MISR swath, and differences in quality control procedures. After applying our bias correction procedures,

Journal of Geophysical Research: Atmospheres
the upper bound estimates of Re w_correct2.1 is now much more consistent with POLDER Re (mean difference of approximately −0.77 μm), even though in some regions (particularly in the Southern Ocean) POLDER Re have slightly larger values than Re w_correct2.1 ; further limiting the POLDER Re to only include grids with standard error of the mean <1.5 μm greatly reduces such differences. It is encouraging that the Re maps in the left column of Figure 5 come from independent retrieval techniques (i.e., polarization, multi-angle,

Journal of Geophysical Research: Atmospheres
and bias correction), and with different spectral channels (i.e., 1.6, 2.1, and 3.7 μm), yet produce consistent results. Spectral channel differences are discussed in more detail in section 3.3.
As a final means of comparison, we turn to global models. Ban-Weiss et al. (2014) showed that MODIS Re 2.1 over oceanic regions are globally~6 μm higher than simulated Re from several General Circulation Models (GCMs). While large differences exist in regional patterns of Re among the GCMs, none match the regional patterns of MODIS Re.

Re w_correct Channel Differences
Platnick (2000) showed that the retrieved Re by the bispectral method is a convolution of the vertically varying Re and a radiative weighting function that is spectrally dependent. The retrieved Re 3.7 is more strongly weighted toward Re values that are nearer to cloud top as compared to Re 2.1 or Re 1.6 because of stronger absorption by liquid water at 3.7 μm. Using a wide range of idealized vertical profiles of Re, Platnick (2000) showed that the influence of vertical variations in cloud microphysics may result in Re differences of up to 1.5 μm among these three spectral channels. This has led to the possibility of retrieving the vertical profile of Re using multiple channels (e.g., Chang & Li, 2002Chen et al., 2008). However, 3-D effects also impact the channel retrievals to various degrees. For example, Zhang et al. (2012) concluded that cloud horizontal heterogeneity can lead to substantial bias between Re 2.1 and Re 3.7 (up to~10 μm for highly heterogeneous cumulus clouds), making the interpretation of vertical variability of Re difficult to ascertain. Figure 6 shows the difference between the mean MODIS  (2011), Figure 6a shows substantial ΔRe 3.7-2.1 differences between the two channel retrievals, ranging from approximately -6 μm in the more cumuliform cloud regions to approximately -1 μm in more stratiform cloudy regions. This range drops substantially for ΔRe UB_correct3.7-2.1 : approximately -2 μm in the cumuliform cloudy regions to slightly positive (up to +0.6 μm) in the stratiform cloudy regions, demonstrating that the bias-correction procedure effectively reduces the disparity between different MODIS spectral Re retrievals to values that are more in line with the simulated values given in Platnick (2000). ΔRe LB_correct3.7-2.1 (Figure 6c) shares the same pattern as ΔRe UB_correct3.7-2.1 , with slightly lower magnitude of approximately −1.2 μm in the cumuliform cloudy regions. The standard error (not shown) associated with ΔRe UB_correct3.7-2.1 and ΔRe LB_correct3.7-2.1 are generally within~0.2 μm in the homogeneous cloud regions and are within~0.6 μm in the heterogeneous cloud regions, with a few latitudes reaching~1.5 μm due to low sampling.
It is evident from Figure 6 that even after the bias correction procedure, the spatial patterns of ΔRe 3.7-2.1 are well preserved in ΔRe UB_correct3.7-2.1 and ΔRe LB_correct3.7-2.1 and that they are dominated by the spatial variability of cloud type (e.g., Stubenrauch et al., 2012).

Conclusion
We employed the method of Liang et al. (2015) to estimate upper and lower bound estimates in the bias of MODIS Re of marine liquid clouds but stratified by SZA, H σ , and τ, to produce bias-corrected Re at regional scales. Using 8 years of January and July MISR and MODIS fusion data, we reported lower and upper bound estimates of bias-corrected mean MODIS Re 2.1 ranging from~4 to 13 μm and~5 to 17 μm, respectively, depending on the cloud regime and SZA. Our results compared favorably to existing in situ validation for MODIS Re (e.g., Ahn et al., 2018;Haney, 2013;Painemal & Zuidema, 2011), to surveys of marine Re from aircraft (Miles et al., 2000), to satellite retrievals from POLDER and ATSR in regions sampled by POLDER, and to GCM generated Re reported in Ban-Weiss et al. (2014), in particular, those from the AM3 GCM. The spatial patterns observed in the Re biases are easily understood in terms of our understanding of global cloud type distributions and the impact of 3-D radiative transfer through heterogeneous clouds on 1-D retrievals of cloud properties (as discussed in section 3.1). The spatial patterns also match nicely with the measured deviations of the angular anisotropy of the radiation field sampled from MISR from 1-D radiative transfer solutions reported in Di Girolamo et al. (2010).
Differences in bias-corrected MODIS Re between Re 3.7 and Re 2.1 remain, ranging between −2 and +0.6 μm, compared to the original MODIS differences of −6 to −1 μm in regional monthly means. The spectral channel differences in bias-corrected Re are more in line with the differences in spectral-channel Re simulated by Platnick (2000) arising from vertical variations in Re. Difference between the bias-corrected MODIS Re 3.7 and Re 2.1 showed a clear dependence on cloud regimes, suggesting very different vertical variations of Re near cloud top between cloud regimes.
The bias-corrected MODIS Re presented herein should give way to better estimates of droplet number concentration (N d ; Grosvenor et al., 2018) and liquid water path (LWP; Zhou et al., 2016;Greenwald et al., 2018), since N d and LWP derived from MODIS are actually derived from the MODIS-retrieved Re and τ. Since we have not assessed the bias in τ, we cannot estimate the biases in N d and LWP at this point. Still the fractional contribution of the Re error to LWP and N d is significant, given that LWP~Re and N d~R e -5/2 (e.g., Grosvenor et al., 2018). We anticipate that the bias-corrected Re results may allow for a refined examination of studies on aerosol-cloud interaction (e.g., Costantino & Bréon, 2010;Myhre et al., 2007).
As noted in section 2, we only considered the MODIS Re retrievals of liquid clouds that are within the MISR swath and passing the quality control procedures for MISR and MODIS data fusion. A total of 48.2% and 51.4% of all fully cloudy, valid MODIS retrievals are used in this study for the months of January and July, respectively. While our results are valid for those samples, we have not assessed their climatological representativeness (also true for the MODIS retrievals as discussed in Cho et al., 2015). Still, the climatology constructed from our samples does compare favorably to ATSR and POLDER ( Figure 5), perhaps indicating that sample representativeness may not have a significant impact on the climatology presented herein.