We examine the viewing zenith angle dependent bias (VZA bias) in warm cloud optical thickness (τ) retrieved from a plane-parallel approach applied to fused Moderate Resolution Imaging Spectroradiometer (MODIS) and Multi-angle Imaging SpectroRadiometer (MISR) data for the months of January and July between 2001 and 2008. The near-simultaneous multiple view-angle observations from MISR offers many advantages over previous τ-VZA bias studies: 1) The analysis no longer requires seasonal and latitudinal cloud invariant assumptions, 2) consistent cloudy scene identification with VZA, 3) stratification of VZA-bias with scene characteristics, and 4) a greater range of sun-view geometries. Contrasting results between previous studies are resolved through careful consideration of the relative azimuth angle (RAZ) between sun and view. Relative to nadir-retrieved τ, τ increases in both forward- and backscatter directions with higher value in backscatter directions for solar zenith angle (SZA) < ~40°. For SZA > ~40°, τ increases with increasing VZA in backscatter directions and strongly decreases in forward-scatter directions. For the most oblique views, ~40–100% absolute monthly mean differences relative to nadir-retrieved τ is common. This behavior is strongly tied to the sampled RAZ and explained based on three factors tied to the spatial heterogeneity of clouds. These factors also explain the behavior of the τ-VZA bias when stratified by nadir-retrieved τ and spatial heterogeneity, even in the thin-cloud limit where sun-glint effects are evident. We also observe an underestimation of τ relative to nadir in the rainbow-scattering directions and attribute it to an overestimation of the cloud-drop effective radius retrieved from MODIS due to cloud heterogeneity. There remains a need to quantify the bias in nadir-retrieved τ as a function of SZA and spatial heterogeneity as a step toward providing bias correction over a wide range of sun-view geometries.
 Cloud optical thickness (τ) is a key variable required in climate research [e.g., Schiffer and Rossow, 1985]. It is routinely retrieved from satellite-measured radiance by assuming clouds and imposed radiative boundary conditions to be horizontally homogeneous. This assumption, often referred to as the plane-parallel assumption, reduces the radiative transfer from three dimensions (3D) to one dimension (1D; the vertical direction), thus making the inversion from satellite-measured radiances to τ tangible along with other assumptions (e.g., the vertical homogeneity of the cloud layer). Because τ is defined as the volume extinction coefficient integrated over the geometric thickness of clouds, its value at a particular location and time should be independent of solar and view geometries under which retrievals are conducted. Numerous studies, however, have shown that satellite-retrieved τ do carry systematic errors that depend on solar and view geometries, thus limiting the utility of satellite-retrieved τ in studying the energy and water cycles within our climate system [e.g., Di Girolamo et al., 2010]. These systematic errors have been traced to two key issues, namely, the nonlinear relationship between τ and bidirectional reflectance factor (BRF) applied to an area-averaged radiance measured within a satellite instantaneous field of view (IFOV) over which clouds are horizontally heterogeneous [e.g., Harshvardhan and Randall, 1985; Marshak et al., 2006; Zinner and Mayer, 2006], and 3D radiative transfer effects arising from the horizontal heterogeneity of clouds [e.g., Davies, 1978; Welch and Wielicki, 1984; Kobayashi, 1993; Barker, 1994; Loeb and Davies, 1996, 1997; Loeb and Coakley, 1998; Várnai and Marshak, 2003; Kato et al., 2006; Várnai and Marshak, 2007]. The following three points summarize the key findings on these issues:
 When the satellite retrieval of τ is performed at nadir for overhead sun, 2D and 3D Monte Carlo radiative transfer simulations through horizontally heterogeneous cloud fields have shown that the retrieved τ will be lower than the truth [e.g., Kobayashi, 1993; Zuidema and Evans, 1998; Kato et al., 2006; Kato and Marshak, 2009]. The underestimation was mainly attributed to the horizontal leakage of radiation from cloud-sides, in addition to the nonlinear relationship between τ and BRF whenever such relationship is of a concave shape.
 Such τ underestimation is reduced to some extent when the satellite retrieval of τ is performed at nadir for moderately oblique sun, but an overestimation in τ could happen for very oblique sun. This was observed by satellite observations [e.g., Loeb and Davies, 1996, 1997; Loeb and Coakley, 1998] and confirmed by 2D and 3D Monte Carlo radiative transfer model simulations [e.g., Zuidema and Evans, 1998; Várnai and Davies, 1999; Kato et al., 2006], which have shown that the retrieved τ increases with increasing solar zenith angle (SZA). It occurs largely because cloud sides have a greater opportunity to intercept more solar radiation for larger SZAs, leading to greater radiance leaving cloud top [Loeb et al., 1997] and a reduction of horizontal leakage of radiation from cloud sides relative to overhead sun [Fu et al., 2000].
 It becomes more complicated when τ is retrieved in an oblique view under oblique sun. Using observations of marine stratus from Advanced Very High Resolution Radiometer (AVHRR) observations, Loeb and Coakley [1998, hereafter referred to as LC98] reported that τ decreases with increasing view zenith angle (VZA) by more than 40% relative to nadir in forward-scatter directions (i.e., for relative azimuth angles between solar and viewing directions within 0°–90°) for SZA > ~50°, while in backscatter directions (i.e., for relative azimuth angles between solar and viewing directions within 90°–180°), τ increases marginally with view angle. These observed behaviors were later supported by 3D radiative transfer simulations [e.g., Loeb et al., 1998; Kato and Marshak, 2009]. In contrast, Várnai and Marshak, 2007 [hereafter referred to as VM07] used Moderate Resolution Imaging Spectroradiometer (MODIS) observations and showed that the retrieved τ increases with viewing obliquity in both forward- and backscatter directions: τ can be more than 40% higher at VZA = 60°, compared to VZA = 0° when SZA >60°. The authors suggested that the likely reasons for the increase in τ with view angle lie in 3D radiative transfer effects and the viewing of cloud sides with relative azimuth angle ranging from 60° to 70° (110° to 120°) in forward-scatter (backscatter) direction, and pointed out that similar behaviors were found in model simulations [e.g., Davies, 1984; Bréon, 1992; Kobayashi, 1993]. Adding to these contrasting results were 3D radiative transfer simulations by Kato et al. , showing that the scene-averaged τ decreases in both forward- and backscatter directions for all azimuth angles; however, no explanation was given for this behavior.
 These contrasting results clearly suggest that the observed bias in retrieved τ with the plane-parallel assumption for real clouds is a complex function of sun-view geometry and cloud heterogeneity that is not fully understood. Assumptions in cloud microphysics may be at play, particularly for comparisons in the backscatter directions [Buriez et al., 2001]. Different data sources and sampling differences may also be at play. For example, Loeb and Davies  examined SZA dependence of τ with 1-year Earth Radiation Budget Satellite (ERBS) observations over ocean between 30°S and 30°N, LC98 examined VZA dependence of τ with 1 month of AVHRR observations of marine stratus off the coasts of California, Peru, and Angola, and VM07 examined the VZA dependence of τ with 1-year global observations from MODIS. Common to these observational studies is the use of wide-swath scanning instruments, resulting in a narrow range of relative azimuth angles and only one view angle observing a given location within the instrument swath. Examining the sun-view bias of retrieved τ with these instruments requires some assumptions: 1) Clouds are diurnally, seasonally, or latitudinally invariant, depending on the orbital configuration, when examining the solar-angle–dependent bias of retrieved τ, 2) clouds are latitudinally invariant when examining the view-angle–dependent bias of retrieved τ from a sun-synchronous orbit, and 3) clouds analyzed in different viewing directions are statistically consistent with respect to cloudy scene identification and the size of the ground IFOV (GIFOV). We will discuss these three assumptions in greater detail in section 2.
 In this study, we overcome many of these assumptions, as detailed in section 2, with near-simultaneous, multiple view-angle observations of clouds from the Multi-angle Imaging SpectroRadiometer (MISR) along with fused observations from MODIS. We present an analysis of SZA-binned view-angle dependence of 1D retrieved τ over the globe based on 8 years of oceanic liquid water cloud observations for the months of January and July, providing ample sampling and seasonal characterization. Our analysis corroborates many of the observed behaviours of 1D retrieved τ measurements with sun-view geometry found in earlier studies and explains the reasons behind some of the contrasting results found in these earlier studies. The near-simultaneous views of the same scene from MISR also allow us to stratify our analysis by cloud optical thickness and spatial heterogeneity and extends the data to a greater range of sun-view geometry yet to be observed or examined.
 The outline of this article is as follows. Section 2 discusses three underlying assumptions for global studies of sun-view dependence of 1D retrieved τ from wide-swath scanning satellite instruments and how MISR observations overcome some of the assumptions for examining view-angle dependence of τ. Sections 3 and 4 describe the data and methodology, respectively, used in our study. Section 5 presents the zonal averaged view-angle dependent bias of 1D retrieved τ from this data stratified by sun-view geometry, cloud optical thickness, and cloud-top heterogeneity, as well as physical insights gained from our analysis. A summary and discussion are given in section 6.
2.1 SZA Dependence
 Some assumptions required for examining the SZA-dependent bias of 1D retrieved τ (hereafter referred to as τ-SZA bias) from a given direction depend on the orbit of the satellite. With a sun-synchronized satellite, the SZAs sampled at a particular location is narrow over the course of 1 month, but can be quite large over the course of 1 year, as demonstrated in Figure 1a for local noon sampling at 30°S. If we set out to determine τ-SZA bias (relative to a specified SZA) at nadir at a particular location at 30°S, we could fit a line to a scatter plot of retrieved τ versus SZA. However, in so doing, we assume that clouds are seasonally invariant, that is, following Figure 1a, the true cloud properties in February (SZA ~10°) are statistically the same as in May (SZA ~40°). If cloud properties do depend on season, then it becomes impossible to decouple the τ-SZA bias from the true underlying changes in the cloud properties with season. Alternatively, we could fit a line to a scatter plot of retrieved τ at a particular VZA versus SZA for a given day by including data over a wide range of latitudes. However, in so doing, we assume that clouds are latitudinally invariant. Also, with a sun-synchronized satellite, the observations of τ at a particular SZA are sampled over the course of 1 year only over a certain range of latitudes. For example, as shown in Figure 1b for noon sampling, observations of τ with SZA = 15° can only be found between 35°N and 35°S, whereas observations of τ with SZA = 60° can only be found within 30°S–80°S and 30°N–80°N. If cloud properties do depend on latitude, then it becomes impossible to decouple the τ-SZA bias from the true underlying changes in the cloud properties with latitude. Because cloud properties do depend on latitude and season, we cannot determine the τ-SZA bias from an instrument in a sun-synchronous orbit. Because the MISR and MODIS instruments used in our study (section 3) are on the Terra satellite platform, which is in a sun-synchronous orbit, we make no attempt at assessing the τ-SZA bias.
 An instrument in a sun-precessing orbit can observe clouds over the same region at various local times (hence different SZAs) over the course of several months. Note that cloud property invariance over this time period and diurnal invariance are still required for assessing the τ-SZA bias. Otherwise, the natural seasonal and diurnal variations are coupled with the τ-SZA bias, and the obtained τ-SZA bias for a location is not particular for a specified season or local time, but for the combination of them. In addition, such examination over the full range of SZA is only possible at low latitudes, as small SZAs would be missing from the analysis at high latitudes. Loeb and Davies  took these issues into consideration when deriving the τ-SZA bias for the ERBS.
2.2 VZA Dependence
 The VZA-dependent bias of 1D retrieved τ (hereafter referred to as τ-VZA bias) has been studied with sun-synchronous, wide-swath scanning instruments, such as MODIS and AVHRR [e.g., LC98; VM07]. The instruments' wide swath provides a moderate range in VZA across the scan that is approximately perpendicular to the orbital direction. Ensembles of retrievals in different VZAs are compared within a narrow range of SZA. However, such an approach to establish the τ-VZA bias assumes cloud properties to be latitudinally invariant. For example, Figure 2 shows the geographical distribution of VZA and SZA of one daytime orbit from Terra-MODIS. We see that a given SZA bin can span a fairly wide range of latitudes (e.g., a SZA = 50°–56° bin spans 45°S–60°S; a SZA = 30–35° bin spans 15°N–30°N). Comparing retrieved τ for different VZA within a SZA bin is also to compare retrieved τ over different latitudes. Thus, τ-VZA bias (relative to a specified VZA) obtained in this way is distorted by cloud latitudinally natural variations, rather than solely indicating a SZA-binned τ-VZA bias.
 The latitudinal invariant assumption can be avoided for instruments in a sun-precessing orbit, like ERBS, by restricting the analysis of τ-VZA bias at a particular latitude and within a given SZA bin. This is achieved by collecting retrievals at one VZA from orbits with one equator-crossing time and at another VZA from orbits with another equator-crossing time when SZAs are the same. However, it may take a long time to achieve significant sampling for the analysis, thus requiring cloud property invariance over long time scales.
 In this study, we avoid the seasonal and latitudinal cloud property invariance assumptions by using data from MISR to examine the τ-VZA bias relative to nadir. This is possible because we can observe the same cloud from multiple viewing directions within minutes of each other (section 4), thus under the same SZA. This also allows us to bin the data by the observed cloud properties at nadir, such as optical thickness and horizontal spatial heterogeneity (section 5).
2.3 Scene Identification and GIFOV Expansion With VZA
 In developing the τ-VZA bias from satellite observations, comparing τ for cloudy scenes from one view to another implicitly requires that the properties of two sets of cloudy scenes be the same. However, one of the limitations in using wide-swath scanning instruments is that cloudy scenes identified across multiple view angles may not be consistent in their properties. For example, a scene classified as partly cloudy at nadir may be classified as fully cloudy in an oblique view [e.g., Minnis, 1989; Zhao and Di Girolamo, 2004]. Even the sensitivity in detecting thin clouds depends on view angle [e.g., Zhao and Di Girolamo, 2004]. These problems are exacerbated by the expansion of the GIFOV with viewing obliquity that occurs for wide-swath scanning instruments. For example, ERBS pixels expand from ~1500 km2 in near-nadir directions to ~82,500 km2 at the limb [Loeb and Davies, 1996, 1997]; MODIS pixels expand from ~1 km2 in near-nadir directions to ~9.6 km2 for VZA = 55° [Nishihama et al., 1997]; and the Global Area Coverage pixels of AVHRR expand from ~4.4 km2 in near-nadir directions to ~52.8 km2 for VZA = ~68° [LC98]. Additionally, the 1D retrieved τ is also a function of the size of the pixel's GIFOV when clouds are not plane parallel. Several studies [e.g., Zuidema and Evans, 1998; Várnai and Marshak, 2001] have shown that the retrieved τ decreases as the pixel size increases from several hundred meters to several tens of kilometers. That the angular distribution of the upwelling radiation field depends on the GIFOV and need not obey directional reciprocity [Di Girolamo et al., 1998] also complicates the development and understanding of τ-VZA bias for instruments with expanding GIFOV with viewing obliquity.
 These problems, however, are minimal in our study. In this study, the cloudy scenes for analysis in multiple views are solely classified based on the nadir view. The multiple views from MISR are registered to the clouds tops viewed at nadir so that cloudy scenes are consistent in different views (section 4). In addition, the GIFOV of MISR expands very little with view angle due to the instrument design. As shown in Zhao and Di Girolamo , the GIFOV for averaging 4 × 4 pixels in native resolution expands from 1.1 km × 1.1 km (~1.2 km2) at nadir to 1.1km × 1.53 km (~1.7 km2) at its most oblique view (70.5°), where the expansion only takes place in the along-track orbital direction. The τ-retrieval bias caused by such a small expansion can be ignored based on Várnai and Marshak [2001, Figure 5 therein].
 MISR, onboard the NASA satellite platform Terra, provides 9 views of the same scene on the Earth within 7 minutes from its multi-camera pushbroom design. Details of the instrument and its performance are described in Diner et al. [1998, 2002]. In brief, one camera views at nadir (nominal VZA of 0°) and is designated AN. Four cameras, designated AF, BF, CF, and DF, point forward along the orbital track at VZAs of 26.1°, 45.6°, 60.0°, and 70.5°, respectively. Four other cameras, designated AA, BA, CA, and DA, point aft along the orbital track at VZAs of 26.1°, 45.6°, 60.0°, and 70.5°, respectively. Radiances are measured in four narrow-band spectral channels (446.4, 557.5, 671.7, and 866.4 nm; Kahn et al. ), with a ground-sampling resolution varying from 275 m to 1.1 km, depending on the channel and camera. Sampling is done over a swath of ~400 km.
 Version 24 of the MISR 866.4 nm radiances are converted to BRFs using equation (1):
where L866 is the radiance measured by MISR and F0 is the solar irradiance corresponding to the MISR 866.4 nm channel (as reported in the MISR radiance file) corrected to the sun-Earth distance at the top of the atmosphere. The sun-view geometries at 17.6 km resolution are linearly interpolated to 1.1 km resolution from Version 13 of the MISR Geometric Parameters product. Only clouds over ocean are considered, where ocean is identified based on Version 24 of the MISR Ancillary Geographic Product (AGP); both deep and shallow ocean categories are included. The AGP file also provides the latitude and longitude for each 1.1 km Space Oblique Mercator (SOM) grid on the World Geodetic System 1984 ellipsoid surface. Clouds over areas prone to sea ice are excluded based on the sea-ice flag in Version 3 of the MISR Terrestrial Atmosphere and Surface Climatology data.
 MODIS [Barnes et al., 1998], also onboard Terra, measures radiance in 36 spectral channels, ranging in wavelength from 0.4 to 14.4 µm over a swath of ~2300 km. The ground-sampling resolution ranges from 250 m to 1 km in the near-nadir directions, depending on the spectral channel, and cloud properties, such as τ and the effective radii (re), are calculated at 1 km resolution. The central (i.e., near-nadir) portion of the MODIS swath completely overlaps with the MISR swath. The cloud re and τ are retrieved through daytime multispectral reflected solar radiances [Platnick et al., 2003]. In this study, cloud properties, including re and cloud phase (MOD06 product), are used to characterize the cloud properties. Only liquid water clouds were considered based on the cloud phase flag. Latitude and longitude (MOD03 product) are used for projecting MODIS data at 1 km resolution to the MISR SOM grid at 1.1 km resolution (section 4). All MODIS products used in this study are from Collection 5 data.
 Terra is in a sun-synchronous orbit with an equator-crossing time of ~10:30 AM local standard time in the descending branch of its orbit. As such, the range of SZA and solar azimuth angle observed at a particular location on Earth by MISR and MODIS within a month are narrow. We use MISR and MODIS data collected over the globe in January and July between 2001 and 2008 in order to sample the boundaries of this range, as well as to capture seasonal differences in cloud cover.
 Given a cloudy scene, we fuse the cloud optical properties retrieved from MODIS and the BRF measurements from MISR. This is achieved by registering the cloudy scene in multiple images of MISR and MODIS with the method described in full in Liang et al.  and further applied in Di Girolamo et al. . In brief, the MODIS retrieved re and cloud phase are first projected on the MISR SOM grid at 1.1 km resolution with General Cartographic Transformation Package routines [U.S. Geological Survey, National Mapping Division, 1993]. The projection is performed by first transferring the latitude and longitude of MODIS pixels to SOM coordinates and then on the MISR SOM grids. The MISR 1.1 km AN-camera image is divided into 3 × 3-pixel domains. If all 1.1 km pixels in a 3 × 3-pixel domain have MODIS re retrievals and are flagged as liquid water phase, then the domain is registered in the MISR oblique camera images to obtain its MISR BRF measurements; otherwise, the 3 × 3-pixel domain is excluded from the analysis (Figure 3). To minimize the registration errors, registration is performed based on the 3 × 3 pixel domains, rather than on individual cloudy 1.1 km pixels. As discussed in Liang et al. , not all cloudy scenes are registered with the same reliability across all cameras. The registration becomes more difficult with viewing obliquity, largely because image texture changes with viewing obliquity. The registration quality-control procedure of Liang et al.  is applied to all nine MISR cameras. The 3 × 3-pixel domains that fail to pass this procedure are excluded from the analysis (Figure 3). Overall, this results in 48.2 and 51.4% of all fully cloudy domains that are flagged as liquid water phase to be registered as a complete set of all nine cameras for the months of January and July, respectively. It is these domains that we include in our analysis, and the quantitative results presented in section 5 should be interpreted with reference to these samples. The impact of this conditional sampling of the domains on our results is discussed in section 6. The impact of the “clear sky restoral” used within MODIS Collection 5, and how they may differ with the upcoming Collection 6, is also discussed in section 6.
 As in Liang et al. , τ is retrieved with MISR-measured 1.1 km resolution 866.4 nm BRF and MODIS-retrieved re on the MISR SOM 1.1 km grid using a look-up table approach. However, unlike Liang et al. , which used the look-up table for the standard MODIS cloud microphysical retrievals (MOD06), here we reconstruct the look-up table so as to extend our retrieval coverage to the complete set of sun-view geometries observed by oblique MISR cameras. The look-up table is calculated by the radiative transfer code, DISORT [Stamnes et al., 1988], using all the same assumptions used by MOD06 over ocean. We set the maximum allowable retrieved τ at 200 to accommodate the greater range and higher bias of τ retrieved from oblique views. The low radiometric noise and large radiometric bit depth of the MISR instrument [Diner et al., 2002] avoids saturation issues caused by the asymptotic behavior of BRF with τ under the sun-view geometries sampled by MISR. We have compared our look-up table to the MOD06 look-up table and they are in excellent agreement for overlapping sun-view geometries, and the relative differences in BRF are mostly within 1%.
 We bin the data into a series of narrow zonal bands with a 2.5° latitude width in the months of January and July, respectively. The narrow latitude range leads to a narrow SZA range for 1-month-long observations that reduces the impact of SZA variation on examining the τ-VZA bias at a particular latitude. Taking the observations in January as an example, the difference between the largest and smallest SZA within any 2.5° latitude bin ranges from 4.6° to 13.5°, with a median value of 9.1°.
 Examining the τ-VZA bias also requires considering the azimuth angles sampled by MISR. Figures 4 and 5 show example characteristics of the azimuth angle of MISR cameras at particular locations and time. Figure 4 gives the typical azimuth ranges of the sun and MISR cameras observed across the MISR swath at three representative latitudes. Except for the AN-camera, the viewing directions are close to the solar plane of incidence in both forward- and backscatter directions at high latitudes; at low latitudes, they are close to the normal of the solar plane of incidence (i.e., more side scatter). Also note that the range of MISR azimuth angles become narrower for more oblique cameras, and that the azimuth angle range for an oblique camera completely lies within the azimuth angle range of the lesser oblique cameras. Also, at high latitudes in the Northern Hemisphere, all cameras pointing forward along the orbital track take observations in the forward-scatter directions and all cameras pointing aftward along the orbital track take observations in the backscatter directions the opposite is true at high latitudes in the Southern Hemisphere. Thus, the data are naturally divided into 10 camera-view bins within each latitude bin: one bin for each of the five MISR cameras (AN-camera, A-, B-, C, and D-cameras) that measure forward-scattered radiance and one for each of the five MISR cameras (AN-camera, A-, B-, C-, and D-cameras) that measure backscattered radiance. Figure 5 shows the range of relative azimuth angle (RAZ) between viewing and solar direction for each MISR camera view as a function of latitude, in the forward- and backscatter directions for January and July. It also shows that RAZ becomes narrower for larger viewing obliquity.
 Given the sampling characteristics of MISR, we only retain cloudy pixels observed in oblique camera views within the RAZ range of the D-camera, separately for the forward- and backscatter directions. This is done to maintain an analysis of the τ-VZA bias for a given month and latitude bin that is not impacted by potential effects of changing RAZ with VZA. This RAZ criterion, however, is not applied to the AN-camera, because the AN-camera observes clouds in a completely different azimuth angle range relative to the oblique cameras (as shown in Figures 4 and 5), given that the center of its swath is at nadir. Fortunately, because τ retrievals are largely insensitive to RAZ in the near-nadir directions, the exclusion of the RAZ criterion to the AN-camera should have little impact on our analysis of the τ-VZA bias.
 For a given latitude and camera-view bin, the τ mean in a day is designated with a standard deviation (SD) , where i is the index of day of month and c is the index of camera-view bin. From a set of the daily τ means in 8 years, the monthly mean for January and July, respectively, is given as shown in equation (2):
where is the total number of all daily τ means in January or July. Given , the SD for all daily means of τ and following the Student's t distribution, the 95% confidence interval (CI) of is estimated by equation (3):
to represent the error in the retrieved , where is the critical value of the Student's t distribution with degrees of freedom. In addition, the mean of the daily in January or July is estimated by equation (4):
 Taking the τ value for MISR-nadir camera as a reference, the monthly mean τ-VZA bias is quantified by equation (5):
and is calculated in the forward and backscatter directions, separately.
 Because the distribution of the observed τ in a day within a latitude bin for a camera view is not necessarily a normal distribution, we also estimate the τ-VZA bias in terms of median value of τ. Similarly, for a given latitude and camera-view bin, the τ median in a day is designated . From a set of the daily τ medians in 8 years, the monthly mean for January and July, respectively, is given as equation (6):
where is the total number of all daily τ medians in January or July. Given , the SD for all daily median of τ, the 95% CI of is given by equation (7):
to represent the error in the retrieved .
 In addition, we calculate the dispersion of τ observed in a day in terms of median as shown by equation (8):
where is jth observed τ value in day i for the MISR camera view c. Thus, the mean of a set of in January or July is shown in equation (9):
 The monthly mean of τ-VZA bias in median value is quantified by equation (10):
5 τ-VZA Bias Results
5.1 Latitudinal Variations
 Figures 6 and 7 show the VZA dependence of , , , , , , , , , , SZA, and D-camera RAZ as a function of latitude for the months of January and July, respectively. The median τ-values are smaller than mean τ values for a given latitude bin, because the distribution of daily observed τ is skewed toward the smaller τ values. Note that both the mean and median τ values for the MISR AN-camera increases from low to high latitudes in both hemispheres. Given that Terra is in a sun-synchronous orbit, observations at a high (low) latitude are generally associated with a large (small) SZA. It would be erroneous to attribute the increase in 1D retrieved τ toward polar latitudes solely as an increase in true τ based on Figures 6 and 7: As discussed in section 2.1, we cannot decouple the natural variation in true τ as a function of latitude from the SZA-induced bias on the 1D retrieved τ.
 Under small to moderate SZAs (SZA < ~40°), Figures 6 and 7 show only a weak dependence of τ with viewing obliquity that is roughly U shaped about the nadir. It is relatively flat (generally within ±10%) from nadir to 60° and then increases for VZA = ~70° (D-cameras). For example, in January, τ at VZA = ~70° is biased relative to nadir by as much as 8% in forward-scatter directions and 28% in backscatter directions for SZA = 30°–38° within 5°N–7°N. VM07, whose analysis was confined to VZA of 0°–60°, observed a similar behavior. Their analysis was presented somewhat differently than here, where their data were divided into spatially homogeneous and heterogeneous populations (spatial heterogeneity is considered here in section 5.3). Their homogeneous populations showed the flat behavior, whereas their heterogeneous population showed the U-shaped behavior, including slightly higher biases in the backscatter as compared to forward-scatter directions.
 Under oblique sun (SZA > ~40°), τ continually decreases in the forward-scatter directions and continually increases in the backscatter directions, with the forward-scatter decrease in τ being more pronounced than the backscatter increase in τ. This behavior becomes more pronounced with increasing SZA. For example, in July, under SZA = 56°–65° (e.g., at 32.5°S–35°S), at VZA = 70.5° is −69 and 28% in the forward- and backscatter directions, respectively, changing to −86 and 59% under SZA = 65°–74° (e.g., at 42.5°S–45°S). Here, our results are different from VM07, who show an increase in τ with viewing obliquity in both forward- and backscatter directions. Instead, our results are consistent with other satellite observations [e.g., LC98] and model simulations [e.g., Loeb et al., 1998; Kato and Marshak, 2009] taken under oblique sun. Why these differences occur is addressed in section 5.2.
 Another interesting feature shown in Figures 6 and 7 is that τ has negatively biased dips relative to nadir in some backscatter directions at certain latitudes. They are highlighted on the figures with thin black dotted lines. As it turns out, these lines also represent the scattering angle close to the rainbow direction. Similar behaviors have also been reported with the Polarization and Directionality of the Earth's Reflectances (POLDER) observations [Buriez et al., 2001], where the 1D retrieved cloud spherical albedo is negatively biased in the rainbow directions. They suggested that the assumed re value used for cloud spherical albedo retrieval is too large. Recall from section 4 that the re used in our study is from MODIS and that it is applied to all MISR cameras. If cloud heterogeneity leads to an overestimation of re from MODIS, as shown in Marshak et al. , then its application here to retrieve τ would produce a low bias in in the rainbow direction—as observed in Figures 6 and 7.
 Figures 6 and 7 also show that the widths of the daily observed τ distributions, quantified by and , become narrower in forward-scatter directions and larger in backscatter directions. Buriez et al.  showed stronger sensitivity to the assumed re in the retrieval of cloud spherical albedo toward backscatter directions, compared with that toward forward-scatter directions. And from Marshak et al. , the bias in re retrievals from MODIS can be quite broad and is sensitive to the heterogeneity of the cloud field. Therefore, uncertainties in the re retrievals from MODIS also contribute to the observed behavior of and , which appears to be consistent with expectations based on the sensitivity simulations presented in Buriez et al.  and Marshak et al. .
5.2 Understanding τ-VZA-bias
VM07 examined many possible factors that could explain the U-shaped behavior of τ-VZA bias in MODIS as we move from backscatter to forward-scatter directions. The following factors were considered: 1) effects of some of the time and space invariance assumptions discussed here in section 2; 2) inhomogeneous and homogeneous clouds occurring at different altitudes and over different surfaces; 3) uncertainties in cloud phase or cloud altitude; 4) cross-track changes in MODIS pixel size; and 5) the viewing of cloud sides from oblique directions. They concluded that the most likely explanation comes from point 5. They argued that pixels classified as cloudy may be only partly cloudy; hence, they contain dark gaps between clouds when viewed from nadir. These dark gaps get filled in by brighter cloud sides when viewed more obliquely. For brevity, we will refer to this as the “gap factor.”
 However, the U-shaped behavior of the τ-VZA bias observed at all SZA sampled in VM07 was not observed in LC98 and was only observed here for SZA <40°. VM07 did consider the differences with LC98, but concluded that the reasons for the differences remained unclear. Here we examine two additional factors that can contribute to the τ-VZA bias, namely, 1) the effects of systematically varying degrees of concavity with VZA in the nonlinear relationship between τ and BRF and 2) the fraction of illuminated or shadowed cloud sides within the satellite IFOV is a function of VZA, SZA, and RAZ.
 It has been known for some time that the effect of subpixel cloud horizontal heterogeneity leads to an underestimate of retrieved τ due to the concavity of the nonliner relationship between τ and BRF [e.g., Cahalan et al., 1994]. However, it has thus far gone unnoticed that, in establishing the τ-VZA bias, the 1D retrieved τ as a function of VZA is subject to systematic concavity changes of the τ-BRF nonlinear relationship with increasing VZA. Figure 8 shows an example of τ-BRF relationships for VZA = 0° and 60° for a SZA = 60° and RAZ = 30°. Given a cloudy pixel consisting of two equal-size cloudy subpixels with τ1 = 2 and τ2 = 8, respectively, and ignoring 3D radiative transfer effects, the retrieved τ is less than the truth, τtruth, both at nadir and in VZA = 60°. Moreover, is smaller than because BRF changes more concavely with τ in VZA = 60° than that at nadir. The magnitude of this difference is a function of the sun-view geometry and degree of subpixel cloud heterogeneity. Figure 9 shows an example for a half-and-half mixture of two optical thicknesses (τ = 4 and 20) for a range of sun-view geometries. The magnitude of negative bias relative to nadir caused by nonlinearity of τ-BRF alone increases with increasing VZA, which would act to oppose a U-shaped τ-VZA bias. Also, the magnitude of τ-VZA bias increases with SZA and decreases, to a lesser extent, with RAZ toward 90° (180°) from 0° (90°) in forward-scatter (backscatter) directions. For the optical thickness mixture (τ = 4 and 20) in this example, the magnitude of the relative bias ranges from −2 to −12% with viewing obliquity. For a greater range of optical thicknesses from 2 to 22, the relative bias can range from −7 to −40% with viewing obliquity even for a moderate oblique sun (SZA = 57°). For brevity, we refer to this as the “concavity factor.”
 Now consider the role of the fraction of illuminated or shadowed cloud sides within the satellite IFOV as a function of VZA, SZA, and RAZ, which we will refer to as the “bump RAZ factor.” Figure 10 illustrates an isolated cloud with a simple geometry and a single bump, observed away from the solar incident plane and illuminated under an oblique sun. The cloud illustrated completely covers the satellite IFOV. In the forward-scatter direction (Figure 10b), as the cloud is viewed from more oblique viewing directions, the fraction of shadowed cloud side B increases and the fraction of illuminated cloud side A decreases within the satellite IFOV. This leads to a smaller BRF with viewing obliquity than that calculated with the 1D nadir-retrieved τ value, hence a decreasing with increasing VZA. Because this cloud is also horizontally heterogeneous within the IFOV, the effect of the concavity factor also contributes to this decrease in with increasing VZA. Note the dependence of this effect with RAZ: The decrease in with increasing VZA is largest for RAZ = 0° and smallest for RAZ = 90° when discussing forward-scatter directions. So if the cloud is not viewed in the solar incident plane (as illustrated), the reduction in BRF (hence ) is weakened by the increased fraction of illuminated cloud side C and decreased fraction of shadowed cloud side D with viewing obliquity.
 Similarly, as the cloud is viewed in more oblique backscatter directions (Figure 10c), the fraction of the illuminated cloud side A increases and the shadowed cloud side B decreases within the IFOV. This leads to a larger BRF with viewing obliquity than that calculated with the 1D nadir-retrieved τ value, hence an increasing with increasing VZA. The dependence of this effect has a maximum (minimum) effect for RAZ = 180° (90°). The increases in with oblique views are weakened by 1) the decreased fraction of the illuminated cloud side D and the increased fraction of shadowed cloud side C within the IFOV and 2) the concavity factor.
 Figure 11 illustrates how the τ-VZA bias varies with VZA for the three factors discussed above and how they add up to produce the observed behavior of the bias. As illustrated, if the bump RAZ factor and concavity factor are the only two factors affecting the τ-VZA bias, then when adding these two factors we should expect a moderate increase in in the backscatter direction and strong decrease in in the forward-scatter direction with increasing VZA. This behavior should be strongest for RAZ = 0°/180° and diminishes for RAZ = 90°. It should also be stronger for larger SZA. The addition of the principle factor given in VM07, namely, the gap factor, pushes this behavior toward a U shape: The increasing with increasing VZA in the backscatter direction becomes even more pronounced, whereas in the forward-scatter direction it acts to diminish the decrease in with increasing VZA and, if dominant, perhaps even causes to increase with increasing VZA. We postulate that these three factors acting together can explain the behavior of the retrieved τ with VZA observed in VM07, LC98, and here, and that any contrasting results between these studies can be rectified when RAZ is carefully considered.
 In the data set used in our study, RAZ are closest to 90° (i.e., close to side scattering; Figures 6 and 7) for small SZAs. They are comparable to those sampled by VM07, where RAZ sampled in VM07 was ~60° in forward-scatter directions and ~110° in backscatter directions over all SZAs. Here, our observations in Figures 6 and 7 and those from VM07 show a U-shaped behavior in that is slightly asymmetric, with higher biases in the backscatter, compared to forward-scatter. As discussed above, RAZ close to side scattering and small SZA lead to a lower impact of the bump RAZ factor and concavity factor, giving way to the dominance of the gap factor. Because the observations are not at SZA = 0° and RAZ = 90°, the bump RAZ factor is not completely absent, leading to the observed asymmetry in the U shape when added to the gap factor.
 At large SZAs, our sampled RAZ moves closer to the solar incident plane, reaching as close as ~28°/152° (Figures 6 and 7). Our observations in Figures 6 and 7 show a moderate increase in τ in the backscatter directions and strong decrease in τ in the forward-scatter directions with increasing VZA, which, based on our argument, indicates the important roles of the bump RAZ factor and concavity factor as we move toward both larger SZA and RAZ closer to 0°/180°. This behavior is also consistent with the AVHRR observations reported in LC98. While the sampled RAZ distributions were not reported in LC98, we anticipate that they are close to the solar incident plane.
5.3 Stratification by Nadir-View Characteristics
 As discussed in section 2, the MISR-MODIS approach taken here in analyzing τ-VZA bias has several advantages over other observational studies, including the ability to stratify the analysis by scene characteristics based only on the nadir view. We consider two nadir-view scene characteristics: cloud optical thickness, τAN, and the BRF spatial heterogeneity parameter defined in Liang et al. , as shown in equation (11):
here is the 3 × 3-1.1-km pixel domain's mean BRF with an SD of σ, which are calculated from the 275 m resolution 866 nm BRF of the MISR AN-camera.
 Figure 12 shows the VZA dependence of as a function of latitude for July of clouds binned into six τAN bins: 0–2, 2–4, 12–16, 24–28, 36–40, and 48–52. Interestingly, for moderate to large τAN, the behavior with VZA and latitude is about the same for all τAN bins. This translates to larger magnitudes of τ-VZA bias with larger τAN, as represents a fractional change in cloud optical thickness relative to nadir.
 For low values of τAN, is everywhere more positively biased than optically thicker clouds. That is, at any latitude, the shape of with VZA is pushed toward a more pronounced U shape as we move toward optically thin clouds. Even for large SZA in the forward-scatter direction, the for optically thin clouds tends to be smaller negative values with increasing VZA, compared with optically thick clouds. This push toward a more U-shaped τ-VZA bias is consistent with our expectations based on our discussion on Figure 11. Small values of retrieved τAN are expected for optically thin, homogeneous clouds. But they are also expected for subpixel clouds. Subpixel clouds are more heavily influenced by the gap factor, which pushes the τ-VZA bias toward a U shape (Figure 11). If the fraction of retrievals containing subpixel clouds increases with decreasing τAN, then we would expect the gap factor to play a larger role for smaller τAN.
 While Figure 12 shows the is everywhere more positively biased for optically thinner clouds, compared with thicker clouds, there is a clear asymmetry in the relative increase, namely, the relative increase from thick to thin cloud in is larger in the forward-scatter direction, compared to the backscatter direction. This is easily explained by an additional factor that influences the τ-VZA bias in the thin cloud limit: sun glint. The MODIS algorithm assumes a 5% Lambert reflecting ocean surface [Meyer and Platnick, 2010]; if a large fraction of the reflected radiance from the ocean surface comes from the solar specular direction, then large deviations from a 5% Lambertian surface would occur. Sun glint impacts τ retrievals for both thin, homogeneous clouds and subpixel clouds. The presence of sun glint would produce a positive bias in the τ retrievals at VZA and RAZ that are about the specular direction. Because the wind-blown ocean is not a smooth surface, glint appears over a wide range of angles about the specular direction. Figure 12 includes the location of scattering angles that are within a 40° cone of the specular direction. The largest increases in from thick to thin cloud fall within this cone. For example, the maximum increase occurs at 7.5°S–10°S for the A-forward-camera, with a value of (τ = 0–2) – (τ = 48–52) = 106%. Note that the large increases do not occur for the large SZA and VZA. For these large angles, even optically thin or broken clouds can produce large extinction along the solar/specular path, thus creating more scattering to space and a more diffusive radiation field at the surface. The more diffusive field and smaller intensities reaching the surface reduce the sun-glint effect at these large angles.
 Figure 13 shows four of the τAN bins in Figure 12, namely, 0–2, 2–4, 12–16, and 36–40, stratified into three Hσ bins: <0.1, 0.1–0.2, and >0.2. As clouds become more spatially heterogeneous (i.e., as Hσ increases) we should expect all three factors, namely, bump RAZ, gap, and concavity factors, to take on a larger role in controlling the τ-VZA bias. For optically thick clouds, we note in Figure 13 a general trend of decreasing in value as the cloud becomes more heterogeneous in forward-scatter directions and increasing in backscatter directions, and that the magnitude of the forward-scatter decreasing is larger than the backscatter increasing. This suggests that the concavity and the bump RAZ factors increase more rapidly than the gap factor with increasing cloud heterogeneity at large τAN in the forward-scatter direction, while concavity factor dominates the bump RAZ and gap factors in the backscatter direction. The lesser importance of the gap factor stands to reason because large values of retrieved τAN are less likely to have subpixel clear gaps, which produces lower values of retrieved τAN. As the clouds become optically thinner, Figure 13 shows that the τ-VZA bias moves toward a U-shaped distribution, indicating a strong influence of the gap factor for smaller retrieved τAN, which is most clearly shown for Hσ > 0.2. Even for smooth clouds (Hσ < 0.1), the influence of the gap factor in pushing the τ-VZA bias toward a more U-shaped distribution as τAN decreases is evident, although to a much lesser extent than for Hσ > 0.2. The greater influence of sun glint at lower τAN is also evident in Figure 13 for all Hσ. While the gap factor does appear to push the τ-VZA bias toward a more U-shaped distribution as τAN decreases, we note that the bump RAZ factor remains influential even for very optically thin clouds, as is evident from the observations in the forward-scatter directions near the solar plane of incidence (i.e., at high latitudes in our data set), where still decreases with increasing VZA.
6 Summary and Discussion
 This study examined the VZA-dependent bias of satellite remotely sensed cloud optical thickness (τ) with the standard plane-parallel assumption. To carry out this study, we fused the multiple view-angle observations of oceanic liquid water clouds from MISR and the observations from MODIS for the months of January and July between 2001 and 2008. The unique, near-simultaneous, multiangular observations from MISR allow us to overcome many shortcomings found in previous observational studies on τ-VZA bias derived from wide-swath, single-view scanning instruments. For example, unlike previous studies, we are able to exclude cloud seasonal and latitudinal invariant assumptions discussed in detail in section 2. Furthermore, the ability to maintain consistency of cloudy scene identification across multiple view angles from MISR has several advantages (discussed in section 2.3) over wide-swath instruments for studying τ-VZA bias. It also enabled us to stratify the τ-VZA bias analysis based on scene characteristics, such as cloud optical thickness and spatial heterogeneity, obtained from one viewing direction. Our study also expands upon the range of observed sun-view geometries sampled in other studies.
 The τ-VZA biases are analyzed in terms of the zonal means of cloud optical thickness retrieved from multiple view directions relative to nadir. Our analysis qualitatively confirmed many τ-VZA biases found in previous observational studies (LC98 with the AVHRR observations, VM07 with the MODIS observations, and Buriez et al.  with POLDER observations). While LC98 report a decrease in τ relative to nadir with increasing viewing obliquity in the forward-scatter directions, and VM07 just the opposite, we observed that both behaviors are possible and depend on the RAZ sampled.
 To explain the behavior of τ as a function of view angle, VM07 argued that the behavior arises because many pixels classified as cloudy may be only partly cloudy; hence, they contain dark gaps between clouds when viewed from nadir. These dark gaps get filled in by brighter cloud sides when viewed more obliquely. While this gap factor is plausible, it alone cannot explain the decreasing τ with increasing viewing obliquity in forward-scatter directions observed here for RAZ close to the solar plane of incident and in LC98. We considered two additional factors, namely, a concavity factor, whereby the effect of systematically varying degrees of concavity with VZA in the nonlinear relationship between τ and BRF induces a VZA-dependent bias on τ, and a bump RAZ factor, whereby the dependence of the fraction of illuminated or shadowed cloud sides within the satellite IFOV on VZA, SZA, and RAZ induces VZA-dependent bias on τ that depends on SZA and RAZ. The combination of these three factors seems to explain the τ-VZA bias over the entire range of sun-view geometries observed here and in VM07 and LC98. They also seem to explain the observed behavior of the τ-VZA bias dependence on nadir-retrieved τ and cloud spatial heterogeneity characterized here by Hσ. In the thin-cloud limit, the impact of the assumed ocean reflectance (5% isotropic reflectance) on the observed τ-VZA bias was evident at scattering angles normally associated with sun glint (i.e., within a 40° cone about the specular direction).
 We also took note of two other observations: 1) The behavior of τ with VZA indicated an underestimate in τ relative to nadir in the rainbow-scattering directions, and 2) the widths of the daily observed τ distributions were narrower in the forward-scatter directions relative to the backscatter directions. We argued in section 5.1, with the aid of findings from Buriez et al. , that these are a strong indication that the cloud drop effective radii (re) retrieved from MODIS is overestimated—a result that is consistent with the effect of cloud horizontal heterogeneity on MODIS-retrieved re [e.g., Marshak et al., 2006]. The magnitude of the bias is not ascertained here.
 Overall, for the RAZs sampled in our data set, we found that τ increases in both forward- and backscatter directions relative to nadir-retrieved τ, with higher value in backscatter directions, for SZA < ~40°. For SZA > ~40°, τ increases with increasing VZA in backscatter directions and strongly decreases in forward-scatter directions. For the most oblique views sampled (VZA = ~70°), ~40–100% absolute monthly mean differences relative to nadir-retrieved τ was common. We stress that these quantitative values are only for those cloudy 3 × 3-pixel domains included in our analysis (section 4). As discussed in Di Girolamo et al. , such quantitative values may not be representative of all 1.1 km oceanic liquid water cloudy pixels. Recall from section 4 that approximately half the fully cloudy 3 × 3-pixel domains were excluded from our analysis. However, the analysis presented in Liang et al.  suggests that this will have a <1% impact on the τ-VZA bias reported here. The impact is expected to be more important for the partly cloudy 3 × 3–1.1 km-pixel domains excluded from our analysis and for those pixels that have been removed from analysis due to Collection 5's use of “clear sky restoral,” whereby cloudy pixels lying at the edge of detectable cloud fields and 1 km pixels appearing as partly cloudy according 250 m resolution channels are restored to “clear sky.” This is because they are likely to be strongly affected by 3D radiative effects. Based on the discussion in section 5.3 of the impact of cloud homogeneity on τ-VZA bias, we may expect that results for the full set of clouds to be more like that of clouds with higher Hσ values. Note, however, that the various factors that contribute to the τ-VZA bias that we examined here apply equally to the full set of clouds.
 The upcoming Collection 6 for MODIS cloud optical properties will remove the process of clear sky restoral. We expect very little change to the τ-VZA biases given in Figures 6 and 7. This is because the pixels affected by clear sky restoral are pixels lying on the edge of clouds and small clouds <1 km. These pixels would have been removed from our analysis due to the strict criteria given in section 4 that leads to fully cloudy 3 × 3-pixel domains. Collection 6 will also replace the Lambertian surface albedo with a Cox and Munk  representation of wind-driven ocean waves. This should greatly reduce the sun-glint artifacts observed in Figures 12 and 13.
 Quantifying the τ-VZA bias as a function of SZA, RAZ, scene characteristics (e.g., nadir-retrieved τ and Hσ), and scale is an important step toward removing bias in τ retrieved from passive satellite sensors caused by algorithm assumptions (e.g., plane-parallel clouds). However, the bias examined here and elsewhere has been relative to the τ retrieved at nadir. There remains a need to quantify the bias in τ retrieved at nadir, which, too, depends on SZA, RAZ, scene characteristics, and scale. Until then, caution is recommended following Di Girolamo et al.  in using retrieved cloud properties derived from passive satellite sensors for studying the spatial and temporal properties of cloud microphysical and optical properties over the globe.
 This work was supported by a NASA Earth and Space Science Fellowship and a NASA New Investigator Program in Earth Science award, both under Program Manager Ming-Ying Wei. Additional support from the MISR project through the Jet Propulsion Laboratory of the California Institute of Technology is also gratefully acknowledged. The authors thank Michael Garay and two anonymous reviewers for their helpful comments on this article. The MISR data were obtained from NASA Langley Research Center Atmospheric Sciences Data Center. The MODIS data were obtained through the Level 1 and Atmosphere Archive and Distribution System of NASA Goddard Space Flight Center.