Angular Distribution Models for Top-of-Atmosphere Radiative Flux Estimation from the Clouds and the Earth’s Radiant Energy System Instrument on the Tropical Rainfall Measuring Mission Satellite. Part I: Methodology

CERES cloud mask

To determine the cloud cover over a CERES footprint, the CERES cloud mask (Trepte et al. 1999; Minnis et al. 1999) is applied to all VIRS pixels that lie within a CERES footprint. The cloud mask consists of a series of threshold tests applied to all five VIRS spectral channels during the daytime (θ_o < 78°, where θ_o is the solar zenith angle at the VIRS pixel), and three channels (3.78, 10.8, and 12.0 μm) at night. If the observed radiances deviate significantly from expected clear-sky radiances in at least one of the available channels, a pixel is classified as cloudy. A cloudy pixel can be classified as either glint, “weak” cloud, or “strong” cloud, depending on how much its radiances deviate from the predicted clear-sky radiances. A clear pixel is classified as weak, strong, or aerosol, where “aerosol” can be smoke, dust, ash, oceanic haze, or “other” (e.g., when a combination of aerosols is detected or when algorithms cannot distinguish between two or more aerosol types). Expected clear-sky radiances are determined on a 10′ latitude–longitude grid. Clear-sky albedo maps (Sun-Mack et al. 1999), directional reflectance models, and bidirectional reflectance functions are used to predict expected clear-sky radiances in the 0.63-, 1.6-, and 3.75-μm channels (Minnis et al. 1999). Top-of-atmosphere brightness temperatures at 3.75, 10.8, and 12 μm are determined using surface skin temperatures and atmospheric profiles from numerical weather analyses and empirical spectral surface emissivities (Chen et al. 1999). Surface elevation, vegetation type, and up-to-date snow-coverage maps are also used to determine the expected clear-sky radiances.

The daytime cloud mask involves a three-step analysis of each pixel. The first step is a simple IR test that flags the pixels that are so cold they must be a cloud. Over ocean, this condition occurs if the VIRS 10.8-μm-channel brightness temperature is more than 20°C below the ocean surface skin temperature. For most land surfaces, a pixel is flagged as cloudy if its 10.8-μm-channel brightness temperature is smaller than the temperature at 500 hPa. A temperature corresponding to a lower pressure is used for surface pressures of less than 600 hPa. The second step involves a series of three tests that compare the pixel to a known background or clear-sky value for 0.63-μm reflectance, 10.8-μm brightness temperature, and 3.75–10.8-μm brightness temperature difference. If all three tests unanimously determine the pixel to be clear (cloudy), this pixel is labeled strong clear (cloudy). If one or two tests fail, a series of additional tests that involve the ratio of 1.6-μm to 0.63-μm reflectances and/or the difference between 11-μm and 12-μm brightness temperatures are applied to determine whether a pixel is weak or strong clear/cloud.

The first step in the nighttime cloud mask is identical to the initial daytime brightness temperature test. The next step consists of tests that compare a pixel's brightness temperature and brightness temperature difference (3.75–11 μm) to predetermined clear-sky values. If either test fails to identify the pixel as clear, then a set of additional tests with a different set of thresholds is used to determine whether a pixel is weak clear or is weak/strong cloud (Trepte et al. 1999).

Over very hot land and desert, the VIRS thermal channels may saturate. To avoid misclassifying clear CERES footprints because of saturated VIRS data, the CERES WN filtered radiance is tested for the possibility that the scene may be clear. This test is used when the VIRS thermal radiance in a CERES footprint is flagged as “bad” and the VIRS 0.63-μm channel contains a good radiance. If the CERES WN filtered radiance exceeds a predetermined threshold, the footprint is reclassified as “clear” and a flux is determined from CERES. Otherwise, the scene type is assumed to be “unknown.” The predetermined CERES WN filtered radiance is derived from radiative transfer model simulations in 5° viewing zenith angle increments over a hot desert scene with a dry tropical atmosphere and a surface temperature of 314 K (D. Kratz 2001, personal communication). Between 35°S and 35°N, saturation occurs in less than 0.5% of the observations. Most of these occurrences are for daytime scenes over desert during the summer months.

Aerosol and cloud property retrieval algorithm

Aerosol optical depths from VIRS pixels identified as clear are inferred from 0.63-μm VIRS radiances based on the retrieval algorithm of Ignatov and Stowe (2002). The algorithm uses a single-channel lookup table approach based on radiances computed from the “second simulation of the satellite signal in the solar spectrum” (6S) radiative transfer model (Vermote et al. 1997). Aerosols are assumed to be nonabsorbing and are represented by a lognormal particle size distribution with a modal radius of 0.1 μm and a standard deviation in the logarithm of particle radius of 2.03 μm. These particle size distribution parameters were determined by fitting Mie calculations for a monomodal lognormal size distribution to an empirically derived phase function (Ignatov 1997).

Radiances from VIRS pixels identified as cloudy are analyzed to estimate parameters that characterize the optical and physical properties of the cloud. These parameters include cloud visible optical depth, infrared emissivity, phase, liquid or ice water path, cloud-top pressure, and particle effective size. The algorithm consists of an iterative inversion scheme to determine the cloud properties that, when input to a plane-parallel radiative transfer model, yield the best match to observed radiances at a particular satellite viewing geometry. A detailed description of the retrieval algorithm and initial results is provided in Minnis et al. (1995, 1998, 1999). Cloud-top height and pressure are determined from the retrieved cloud-top temperature using the nearest vertical temperature and pressure profiles from numerical weather analyses. Liquid and ice water paths are derived from retrievals of cloud optical depth and particle effective size.

In cases in which the cloud algorithm cannot determine a solution for the observed radiances, a second cloud mask based on Welch et al. (1992) is used to reassess whether the pixel is really cloudy. The pixel is reclassified as clear if this second cloud mask determines it to be clear. Otherwise, the pixel is labeled as “cloudy no retrieval.” The no-retrieval classification is used for approximately 4% of all cloudy cases.

CERES PSF convolution and cloud layering

Accurate relationships among aerosol, cloud, and radiative fluxes require accurate spatial and temporal matching of imager-derived aerosol and cloud properties with CERES broadband radiation data. When CERES is in cross-track mode, VIRS and CERES observe a scene simultaneously. However, scenes observed by CERES in the along-track direction at oblique viewing zenith angles are observed by VIRS within ≈2 min of CERES. To achieve the closest spatial match between CERES and VIRS, the distribution of energy received at the CERES broadband detectors must be taken into account when averaging imager-derived properties over the CERES footprint. This distribution of energy is described by the CERES point spread function (Smith 1994). The PSF accounts for the effects of detector response, optical FOV, and electronic filters. To determine appropriately weighted and matched aerosol and cloud properties within CERES FOVs, pixel-level imager-derived aerosol and cloud properties are convolved with the CERES PSF.

Within a CERES footprint, the properties of every cloudy imager pixel are assigned to a cloud layer. If there is a significant difference in cloud phase or effective pressure within a CERES FOV, up to two nonoverlapping cloud layers are defined. In general, a single footprint may contain any combination of clear area and one or two distinct cloud areas (Fig. 1).

To determine whether two distinct cloud layers are present, the imager pixels are initially binned by phase into either water or ice categories. Two distinct cloud layers are present if (i) the mean and standard deviation of effective cloud pressure from the two populations are significantly different based on a Student's t test (at the 95% confidence interval) and (ii) the mean cloud effective pressure differs by more than 50 hPa. If both conditions are met, a threshold effective pressure is defined at the midpoint between the effective pressures of the lowest and highest cloud layers. The imager pixels are then recategorized using the threshold effective pressure before the PSF weighted-average cloud properties are determined for each layer.

If this method fails to identify two distinct cloud layers, a second approach is considered. The pixel-level cloud effective pressures are sorted from lowest to highest. The largest gap in this series (exceeding 50 hPa) is used to separate pixels into two cloud layers. The Student's t test is then performed on the mean and standard deviation of the cloud effective pressures for these two populations. If they are statistically different, they are convolved over the footprint as two separate layers. If the pixels fail to meet these minimum requirements, they are assigned to one layer. When present, multilayer imager pixels (e.g., thin cirrus over low cloud) are identified with an overlapped cloud detection algorithm (Baum et al. 1999), but cloud properties are retrieved and convolved as if only one layer were present. The overlapped cloud detection algorithm only identifies multilayer clouds when a well-defined thin upper-level cloud layer lies above a well-defined lower-level cloud (Baum et al. 1999).

Cloud effective parameters over CERES footprints

The cloud fraction over a CERES footprint is determined from 1 − A_clr, where A_clr is the imager clear-area fractional coverage. A cloud fraction is determined only over the part of a CERES footprint that has imager coverage. Footprints near the edge of the VIRS swath have only partial coverage by VIRS. Partial imager coverage can also be due to bad imager data or because a pixel cannot be determined as clear or cloudy by the CERES cloud mask. All full and partial Earth-view CERES FOVs that contain at least one imager pixel are recorded in the SSF product. The effective mean of a parameter x over a CERES footprint is derived from the PSF-weighted layer mean values as follows:

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where A₁ and A₂ are the fractional coverage of layers 1 and 2, respectively, over a CERES footprint.

Under some conditions, a pixel can be identified as cloudy but the cloud algorithm may fail to determine cloud properties from the observed radiances. These cases, referred to as no retrievals, can occur alongside pixels for which the cloud algorithm does provide cloud properties. When this pattern occurs, the region in which retrievals are available is assumed to provide the mean cloud properties over the CERES footprint. That is, we assume that the cloud mean properties over the region of no retrievals are the same as over the region for which retrievals are available.

Because CERES relies on the imager to identify the scene within a footprint, a minimum amount of imager coverage and cloud property information is needed to construct ADMs. The total fraction A_unk of unknown cloud properties over the footprint is determined by combining the imager coverage A_im and the fraction A_ncl of the cloudy area lacking cloud properties as follows:

A_unkA_imA_imA_clrA_ncl(2)

where the first term provides the fraction of the footprint with no imager coverage, and the second term is the fraction of the footprint from the cloudy area with unknown cloud properties. In general, only footprints with A_unk ≤ 0.35are used to construct CERES ADMs. For cloudy scenes over ocean observed at glint angles γ of less than 40°, only footprints with A_im ≥ 0.5 are considered. Here γ is the angle between the reflected ray and the specular ray for a flat ocean given by

View Expanded

where μ and μ_o are the cosine of the viewing and solar zenith angles, respectively, and ϕ is the relative azimuth angle. Over all surfaces except snow, cloudy footprints must have a valid cloud optical depth in the lower layer to be considered. Although footprints with insufficient imager coverage or cloud property information are not considered when constructing the ADMs, a flux estimate is nonetheless provided for these footprints when the ADMs are applied to determine TOA fluxes. The strategy for estimating fluxes from footprints with insufficient imager or cloud property information is described in section 5c.

Interpolation bias correction

To estimate a flux from a radiance measurement, the appropriate ADM scene type must first be determined from the imager retrievals. Next, Eq. (5) is applied using an estimate of the anisotropic factor. However, because the anisotropy of Earth scenes generally varies with viewing geometry and cloud/clear-sky properties in a continuous manner, whereas the CERES ADMs [Eqs. (11) and (12)] are defined for discrete angular bins and scene types, an adjustment to the CERES anisotropic factors is needed to avoid introducing large instantaneous flux errors or sharp flux discontinuities between angular bins or scene types. One way of reducing angular bin discretization errors is to obtain anisotropic factors by linearly interpolating bin-average ADM radiances [e.g., I_j^SW(θ_oi, θ_k, ϕ_l; h_sfc)] and fluxes [e.g., F_j^SW(θ_oi; h_sfc)] to each observation angle (θ_o, θ, ϕ) and evaluating anisotropic factors from Eq. (6) using the interpolated quantities. In addition, interpolation over other parameters that influence anisotropy (e.g., cloud optical depth) can also be used. In some cases it may even be advantageous to combine empirical and theoretical ADMs to estimate the anisotropic factor at a particular angle (e.g., clear ocean SW ADMs in section 6a).

When linear interpolation is used, the instantaneous TOA flux is given by

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where R̃_j(θ_o, θ, ϕ; h_sfc) represents an anisotropic factor at the surface reference level determined from interpolated ADM radiances [Ĩ_j(θ_o, θ, ϕ; h_sfc)] and fluxes [F̃_j(θ_o; h_sfc)]. Although instantaneous flux errors are likely reduced with this approach, there is no guarantee that ensemble averages of the instantaneous fluxes will remain unbiased. A bias in the mean flux will occur if linear interpolation is used when the actual radiance varies nonlinearly within an angular bin. It also occurs when theoretical models are used to supplement empirical ADMs. The bias for a specific scene type j in angular bin (θ_oi, θ_k, ϕ_l) is determined from the difference between the estimated mean flux and the ADM mean flux (determined by direct integration of radiances) as follows:

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where the first term on the right-hand side is the average of all instantaneous flux estimates from Eq. (14) falling in angular bin [θ_oi ± (Δθ_o)/2, θ_k ± (Δθ_k)/2, ϕ_l ± (Δθ_l)/2] for scene type j, and F_j(θ_oi; h_sfc) is the corresponding ADM flux. To remove the bias, a correction term is added to instantaneous TOA fluxes:

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where

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When the ensemble average of instantaneous TOA fluxes from Eq. (16) is determined, the mean flux is unbiased because δF_j(θ_oi, θ_k, ϕ_l; h_sfc) = ΔF_j(θ_oi, θ_k, ϕ_l; h_sfc). This procedure is used in all SW TOA flux estimates (except over snow). In the LW and WN channels, δF_j(θ) is close to zero and is therefore not explicitly accounted for.

TOA flux reference level

Based on theoretical radiative transfer calculations using a model that accounts for spherical Earth geometry, Loeb et al. (2002) recently showed that the optimal reference level for defining TOA fluxes in Earth radiation budget studies is approximately 20 km. This reference level corresponds to the effective radiative “top-of-atmosphere” because the radiation budget equation is equivalent to that for a solid body of a fixed diameter that only reflects and absorbs radiation. The TOA flux at the 20-km reference level h₂₀ is determined from the flux at the surface reference level as follows:

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On the CERES SSF product, instantaneous TOA fluxes are provided only for CERES radiances with θ(h_sfc) ≤ 70° and θ_o ≤ 86.5°.

Footprints with insufficient imager information

As noted in section 3d, CERES footprints sometimes lack sufficient imager information to define an ADM scene type because part of the footprint may lie outside the VIRS imager swath, because the imager data are flagged as bad, or because the cloud algorithm fails to determine cloud properties from the observed radiances (no retrievals). If the total fraction of unknown cloud properties [defined in Eq. (2)] exceeds a threshold, the footprint is not used in ADM development. It is tempting also to reject such footprints when applying the ADMs in determining instantaneous TOA fluxes, but this rejection could introduce systematic biases in the mean TOA flux if the no-retrievals assignments are correlated with cloud type (e.g., thin ice clouds). To avoid introducing potential biases in regional TOA flux estimates caused by systematic rejection of clouds whose optical properties fall outside the expected range of the retrieval model (resulting in no-retrievals assignments), instantaneous TOA fluxes are estimated regardless of what the total fraction of unknown cloud properties is over the footprint. TOA flux estimates for these footprints likely have greater instantaneous errors than those derived with complete imager information, but biases in the overall means will be avoided if the errors are random.

To determine a TOA flux for a footprint that lacks sufficient imager information to define an ADM scene type, ADM radiances [e.g., I_j^SW(θ_oi, θ_k, ϕ_l; h_sfc)] are interpolated to the EOV viewing geometry and are compared with the measured radiance. The anisotropic factor used to convert the measured radiance to flux is evaluated from the ADM whose interpolated radiance most closely matches the measured radiance. To constrain the result, only ADMs having the same underlying surface type as the measurement are considered as possible candidates. From the 9-month CERES–TRMM dataset, footprints with insufficient imager coverage to determine an ADM scene type occur approximately 7% of the time.

Mixed scenes

When a CERES footprint contains a mixture of surface types (e.g., ocean and land, land and desert), instantaneous TOA fluxes are determined using the ADM that corresponds to the surface type with the highest percent coverage over the footprint. For example, near coastlines, if most of the footprint PSF-weighted area is over ocean, an ocean ADM is used to convert the radiance to flux. In converse, if most of the footprint area is over land, one of the land ADMs is used. An exception occurs when SW TOA fluxes are estimated from mixed land–ocean footprints in the sunlight region. In that case, if the glint angle [Eq. (3)] is ≤40° and the footprint is covered by more than 5% ocean, the footprint bidirectional reflectance is assumed to be closer to that for ocean, and one of the ocean ADMs is used.

Clear ocean

Clear footprints are defined as footprints with ≥99.9% of VIRS imager pixels identified as cloud free. Separate clear ocean ADMs are defined for four intervals of wind speed corresponding to the 0–25th, 25th–50th, 50th–75th, and 75th–100th percentiles of the wind speed probability density distribution. These correspond to wind speed intervals of approximately <3.5, 3.5–5.5, 5.5–7.5, and >7.5 m s^–1. The wind speeds, which correspond to the 10-m level, are based on Special Sensor Microwave Imager (SSM/I) retrievals (Goodberlet et al. 1990) that have been ingested into the ECMWF data assimilation analysis. For a given wind speed interval w_j, the ADM is defined following the procedure outlined in section 4.

Because the anisotropy of clear ocean scenes also depends on aerosol optical depth, this dependence should also be accounted for when estimating SW fluxes over clear ocean. The SSF product provides aerosol optical depth retrievals (Ignatov and Stowe 2002), but only in viewing conditions for which the glint angle exceeds 40°. It is consequently not possible to construct empirical ADMs stratified by the Ignatov and Stowe (2002) aerosol optical depth retrievals because no information on how CERES radiances vary with aerosol optical depth in the glint region is available. As an alternative, instantaneous TOA fluxes are first inferred in any viewing geometry from wind speed–dependent empirical ADMs. Next, these TOA flux estimates are adjusted as follows:

View Expanded

where R̃(w_j, θ_o, θ, ϕ; h_sfc) is determined from the wind speed–dependent ADMs, and R^th (w_j, I) and R^th(w_j, Ĩ) are anisotropic factors inferred from the measured CERES radiance I(θ_o, θ, ϕ; h_sfc) and the interpolated ADM radiance Ĩ(θ_o, θ, ϕ; h_sfc), respectively. To determine R^th(w_j, I) and R^th(w_j, Ĩ), CERES radiances I(θ_o, θ, ϕ; h_sfc) and Ĩ(θ_o, θ, ϕ; h_sfc) are compared with lookup tables of theoretical SW radiances stratified by aerosol optical depth. Here, R^th(w_j, I) and R^th(w_j, Ĩ) correspond to the aerosol optical depth for which the theoretical radiances match the CERES radiances. The radiative transfer calculations are based on the DISORT model (Stamnes et al. 1988) and assume maritime tropical aerosols based on Hess et al. (1998) evaluated at 24 optical depths. The ocean surface in the calculations accounts for the bidirectional reflectance of the ocean at the five wind speeds that correspond to the midpoints of the CERES ADM wind speed intervals using the ocean surface bidirectional reflectance “OCEABRDF” subroutine from the 6S radiative transfer code (Vermote et al. 1997). This routine accounts for specular reflection (Cox and Munk 1954), wind speed–dependent whitecaps (Koepke 1984), and below–water surface reflectance (Morel 1988).

Equation (19) can be used to estimate TOA flux in any viewing geometry. However, as the satellite viewing geometry moves towards the ocean specular reflection direction, the radiance increase for a change in angle as small as 1° can be very large. Because such changes are unresolved by the relatively coarse angular bins used to define CERES ADMs, instantaneous TOA flux estimates are generally unreliable for footprints near the specular reflection direction. As a consequence, the radiance-to-flux conversion is not performed in these regions. However, ignoring these samples (e.g., by not providing a TOA flux estimate) can introduce biases in regional mean fluxes because fluxes over cloudy portions of a region will contribute disproportionately to the overall regional mean. To avoid this bias, fluxes in cloud-free sunlight are given by the clear ocean wind speed–dependent ADM flux (F_j^SW) interpolated at the solar zenith angle of the observation.

To determine whether a footprint is too close to the specular reflection direction to provide a reliable flux retrieval, the derivatives of clear ocean ADM anisotropic factors with respect to illumination and viewing geometries (∂R_j/∂θ_o, ∂R_j/∂θ, and ∂R_j/∂ϕ) are evaluated in each CERES angular bin. If an observation falls in an angular bin for which one of the derivatives exceeds a threshold value, a radiance-to-flux conversion is not performed. In this study, a threshold of 0.075 per degree is used as the cutoff, which corresponds approximately to a 40° glint angle threshold.

Clear land and desert

The anisotropy of surface-leaving radiances over land and desert regions depends on several factors, including vegetation coverage, surface type, and surface heterogeneity (Roujean et al. 1992). The intervening atmosphere modifies the surface anisotropy, particularly at shorter wavelengths (Zhou et al. 2001) and for large aerosol optical depth (Li et al. 2000). The observed anisotropy of TOA-leaving radiances also depends on instrument resolution, because clear land scenes become more inhomogeneous when observed at larger spatial scales.

The inclined orbit of the TRMM satellite provides a unique opportunity for determining ADMs under all solar zenith angle conditions. To account for climatological differences between surface types, ADMs are first constructed for each of the International Geosphere Biosphere Programme (IGBP) Global Land Cover types (Loveland and Belward 1997) for which there are sufficient data in the Tropics. CERES uses a 10′ latitude by 10′ longitude map of IGBP types that covers the globe (D. A. Rutan and T. P. Charlock 2001, personal communication). The IGBP classification scheme is provided in Table 1, along with the fraction of cloud-free CERES footprints for each IGBP surface-type category over the entire 9 months of daytime CERES–TRMM observations (last column). Over land and desert, barren desert (16) and open shrubs (7) account for 53% of the clear footprints, IGBP types with low-to-moderate tree/shrub coverage (i.e., IGBP types 9–14) account for 34%, and IGBP types with moderate-to-high tree/shrub coverage (i.e., IGBP types 1–6 and 8) account for 13%. It is unfortunate that there are not enough data over the Tropics to construct ADMs for deciduous needleleaf forests (3), permanent wetlands (11), and urban (13) IGBP types.

Figures 5 and 6 show clear-sky ADM anisotropic factors for θ_o = 30°–40° for individual IGBP types (colored lines) together with ADMs determined by grouping all IGBP types falling in the moderate-to-high and low-to-moderate tree/shrub coverage classes, respectively (solid circles). Of interest is that ADM anisotropic factors for individual IGBP scene types show a remarkable similarity to one another and to the combined low-to-moderate and moderate-to-high tree/shrub coverage classes. In Fig. 5, deviations in anisotropic factors for individual IGBP types from the moderate-to-high tree/shrub ADM class occur primarily in angular bins that are poorly sampled. This result is particularly evident for the closed shrubs (6) and mixed forest (5) IGBP types. The most persistent differences between anisotropic factors from the individual IGBP types and the combined low-to-moderate tree/shrub coverage class occur close to nadir: grassland (10) anisotropic factors are generally larger by up to 4% (relative difference), whereas anisotropic factors for crops (12) are generally lower by up to 3%. Differences also increase at larger viewing zenith angles, at which the data sampling is reduced (note that because the CERES SSF only retains footprints within the VIRS swath, oblique viewing zenith angles are only sampled when CERES is in RAP or along-track mode, which only occurs every third day of data acquisition).

To reduce errors in flux attributable to poorly sampled ADM angular bins, the CERES ADMs are constructed using the low-to-moderate and moderate-to-high tree/shrub coverage classes to determine fluxes over land. For these cases, the variability in the anisotropic factors is estimated to be less than 0.04 at the 95% confidence level for most solar zenith and viewing zenith angle bins. The variability in anisotropic factors is estimated from the variability in daily mean radiances for each angular bin [Eq. (13)].

ADM anisotropic factors for θ_o = 30°–40° for two IGBP types characteristic of desert regions are presented in Fig. 7. Open shrubs (7) are prevalent over west and central Australia, the southwest parts of North America, South America, and Africa, and in central Asia. Barren deserts (16) are associated primarily with the Saharan, Arabian, Thar, and Gobi deserts. As shown in Fig. 7, ADMs are different for these two IGBP types. The ADMs over barren desert regions are more isotropic, presumably because of the lower vegetation coverage there. Capderou (1998) showed similar differences based on ScaRaB measurements from the Meteor-3-07 satellite.

To examine how well the IGBP classification separates the two classes of desert, relative frequency distributions of SW reflectance were determined in each angular bin. Shortwave reflectance is inferred from a measured SW radiance as follows:

View Expanded

where μ_o = cosθ_o, E_o is the incident solar irradiance (=1365 W m^–2), d corresponds to the Earth–Sun distance at the time of observation, and d_o is the mean Earth–Sun distance.

Figure 8 shows results for angular bin θ_o = 40°–50°, θ = 0°–10°, and ϕ = 70°–90°. The two desert types have a well-defined primary peak at reflectances near 15% (open shrubs) and 30% (barren desert), there is a secondary peak in the barren desert distribution near 15%, and there is a hint of a secondary peak in the open shrubs distribution at reflectances near 25%. The reason for the multiple peaks in the two reflectance distributions may be because the fixed IGBP map cannot account for annual or seasonal changes in vegetation type and cover. To provide a better separation between the two desert types, all CERES footprints in 10′ desert regions are reclassified as either “dark” or “bright” desert. Regions with CERES SW reflectances closer to the primary peak of the open shrubs reflectance distribution are classified as dark desert, whereas regions with CERES SW reflectances closer to the primary peak of the barren desert reflectance distribution are classified as bright desert.

The largest difference in ADM characteristics between the reclassified desert ADMs (dark and bright desert) from the original IGBP classes (open shrubs and barren deserts) occurs for the bright desert class. In that case, the bright desert ADMs are more isotropic than the barren desert ADMs, particularly in the forward scattering direction, for which differences in anisotropic factors can reach 6%. In addition, for both desert types, the ADM variability estimate [Eq. (13)] is much smaller for the new dark and bright desert classes. For these cases, the variability in the anisotropic factors is estimated to be less than 0.03 at the 95% confidence level for most solar zenith and viewing zenith angle bins.

Capderou (1998) recently constructed clear desert ADMs using measurements from the ScaRaB instrument on board the Meteor-3-07 satellite. Using scene identification based on the ERBE maximum likelihood estimation (MLE) technique (Wielicki and Green 1989) to identify clear scenes over the Saharan, Arabian, Namib–Kalahari, and Australian deserts, Capderou (1998) derived ADMs for dark and bright desert conditions. To compare the ScaRaB and CERES ADMs, the CERES ADMs are adjusted to the midpoint of the ScaRaB ADM angular bins (ScaRaB uses the same angular bin definition as ERBE) by interpolating CERES ADM mean radiances (I_j^SW)and fluxes (F_j^SW) to the angular bin midpoints and inferring the anisotropic factors from the ratio. The ScaRaB–CERES ADM differences are converted to equivalent SW flux differences by inferring fluxes from the CERES ADM mean radiances (I_j^SW) using both sets of ADMs in each ScaRaB ADM angular bin (θ > 75° excluded) as though the radiances were instantaneous values. Figure 9 shows the resulting SW flux differences and root-mean-square (rms) differences as a function of solar zenith angle inferred from all angular bins. Also provided are results comparing fluxes based on ERBE (Suttles et al. 1988) and CERES desert ADMs. For solar zenith angle bins <60°, the ScaRaB and ERBE fluxes are generally within 3 W m^–2 of the CERES fluxes for both the dark and bright desert models. At larger solar zenith angles, both the ScaRaB and ERBE fluxes are lower than the CERES fluxes by up to 7 W m^–2.

The cause for the increase in flux difference with solar zenith angle is unclear. Further examination of the ScaRaB and ERBE ADMs reveals some large jumps in the nadir anisotropic factors at solar zenith angles >60°. Figures 10a–h show the CERES, ScaRaB, and ERBE ADM anisotropic factors for the θ = 0°–15° ERBE angular bin against solar zenith angle in each of the ERBE relative azimuth angle bins. Whereas the CERES anisotropic factors show a smooth dependence on solar zenith angle, the ScaRaB and ERBE models are much noisier, in particular at the larger solar zenith angles. The cause for the large variability in the ScaRaB and ERBE models may be poor sampling or possibly a solar zenith angle–dependent bias in the MLE scene identification.

Clouds over ocean

The ADM scene-type stratification for clouds over ocean is provided in Table 2. There are two phase categories, 12 cloud-fraction categories, and 14 cloud-optical-depth categories. Phase over a CERES footprint is inferred from VIRS-imager pixel-level phase retrievals (Minnis et al. 1998). Each VIRS-imager pixel within a CERES cloud layer is assigned a phase index of 1 for liquid water and 2 for ice. The pixel-level phase indices are weighted by the CERES PSF to yield the effective phase over each layer. The effective phase over the entire footprint is determined by area-averaging the phase indices of each layer using Eq. (1). ADMs for “liquid clouds” are determined from footprints with an effective phase index <1.5, and ADMs for “ice clouds” are determined from footprints with an effective phase index ≥1.5. A separate class for “mixed-phase” footprints would be desirable, but the sampling is limited, with only 9 months of CERES–TRMM observations.

Although the number of ocean cloud ADM scene types can potentially reach 336 (i.e., 2 × 12 × 14), the actual number of scene types with sufficient data is much lower. In most solar zenith angle bins, 72 (or 43%) of the possible liquid water cloud classes have sufficient data to build an ADM, whereas 57 (or 34%) of the possible ice cloud classes have sufficient sampling. Figures 11a and 11b show the frequency of occurrence of liquid water and ice cloud ADM classes, respectively, by cloud fraction and cloud optical depth. When cloud fraction is low, only the thin ADM cloud classes are sampled. As the cloud fraction increases, the range in cloud optical depth increases, and more cloud-optical-depth ADM classes appear. A similar broadening in cloud-optical-depth distributions with cloud cover was also observed by Barker et al. (1996).

When clouds are present, 37% of the footprints fall in the overcast (99.9%–100%) cloud-fraction class. If only footprints dominated by liquid water clouds (i.e., footprint effective phase index <1.5) are considered, the fraction of overcast footprints drops to 25%, as compared with 72% for footprints dominated by ice clouds (i.e., footprint effective phase index ≥1.5). Overall, the frequency of occurrence for the liquid water cloud class is 75%, as compared with 25% for the ice cloud class.

Examples of ADMs for thin (cloud optical depths of 1.0–2.5) and thick (cloud optical depths of 20–25) ice clouds for θ_o = 50°–60° are provided in Figs. 12a,b. For the thin-cloud case (Fig. 12a), the anisotropic factor ranges from 0.6 to 3.3 as compared with 0.9 to 1.6 for the thick-cloud case. The largest sensitivity to cloud optical depth occurs at near-nadir views, for which the anisotropic factor changes by 50%. Figures 12c,d show differences in anisotropic factors between liquid water and ice cloud ADMs for the same cloud-optical-depth intervals as in Figs. 12a,b. For both the thin-cloud (Fig. 12c) and thick-cloud (Fig. 12d) conditions, anisotropic factors for liquid water clouds exceed those of ice clouds in the forward and backscattering directions but are smaller at relative azimuth angles in the side scattering direction. A similar dependence was observed from POLDER measurements by Loeb et al. (2000). As expected, the magnitude of the differences is smaller for the thicker-cloud case (Fig. 12d) because increased multiple scattering in these clouds reduces the sensitivity to differences in liquid water and ice cloud phase functions. However, because albedos and fluxes are so much larger for the thicker clouds, even small errors in anisotropy can cause large albedo errors. For the cloud cases shown in Fig. 12, ignoring phase would result in mean albedo errors of up to 0.05.

Figure 13 compares CERES ADMs for overcast ice clouds for each of the 14 cloud-optical-depth intervals together with the one ERBE overcast ADM. For this comparison, the CERES ADMs are interpolated to the midpoints of the ERBE angular bins (Suttles et al. 1988). The ERBE overcast model most closely follows the CERES ADM for cloud-optical-depth interval 12.5–15. The ERBE anisotropic factors exceed CERES values by up to 60% for thin clouds near nadir; for viewing zenith angles between 40° and 60°, anisotropic factors are insensitive to cloud optical depth, consistent with theoretical simulations by Davies (1984).

Because of the strong sensitivity in the anisotropic factors to cloud properties (i.e., cloud optical depth and cloud fraction), the ADM lookup tables under cloudy conditions are interpolated not only to the measurement viewing geometry, but also to the effective cloud fraction and cloud optical depth over the footprint. The interpolation procedure is the same as that outlined in section 5a but involves interpolation over two extra variables.

Clouds over land

The ADM classes for cloudy conditions over land are stratified by the four land types considered for clear conditions (section 6b), two cloud-phase classes (defined in the same manner as for clouds over ocean), five cloud-fraction classes and six cloud-optical-depth classes (Table 2). Because only 9 months of CERES–TRMM measurements are available, the number of classes over each of the land types is reduced relative to that over ocean to ensure a sufficient number of samples to construct an ADM.

Snow

Because the TRMM orbit is restricted to 35°S–35°N, sampling under snow conditions is insufficient for developing empirical ADMs. As an alternative, fluxes under snow conditions for CERES–TRMM are determined from theoretical ADMs based on 12-stream DISORT (Stamnes et al. 1988) model calculations. In the calculations, the surface bidirectional reflectance of snow is accounted for explicitly by inserting a packed snow layer at the bottom of the atmosphere (0–1-km altitude). Within the snow layer, ice particles are assumed to be spheres, having a lognormal size distribution with a mode radius of 50 μm and a standard deviation of 2.0 μm. The concentration of ice particles is 1.0 × 10¹² m^–3, which corresponds to a density of ≈0.5 Mg m^–3. Ice refractive indices from Warren (1984) are used to compute the optical properties of ice particles from Mie theory. The atmosphere is divided into six layers. Absorption by water vapor, ozone, carbon dioxide, and oxygen are based on k-distribution tables (Kato et al., 1999) based on a midlatitude summer atmosphere (McClatchey et al. 1972).

Under cloudy conditions, a liquid water cloud layer between 1 and 2 km is inserted above the snow layer in the DISORT model calculations. The cloud particles are also taken to be spheres, having a lognormal distribution with a mode radius of 10 μm and standard deviation of 1.42 μm. The cloud optical depth is fixed at 10. Radiances are computed at 18 solar zenith angles, 51 viewing zenith angles, and 61 relative azimuth angles. TOA fluxes from the theoretical ADMs are determined using Eq. (14).

LW and WN ADM scene-type parameters

Cloud categories are based on the imager-derived cloud fraction over the CERES footprint. A footprint is assumed to be clear when the cloud fraction is ≤0.1%. “broken” when the cloud fraction is between 0.1% and 99.9%, and “overcast” when the cloud fraction is ≥99.9%. In Table 3, ocean is defined by IGBP type 17 (Table 1), desert is defined by IGBP types 7 and 16, and land is all IGBP types except 7, 15, 16, 17, 19, and 20. Over snow, empirically derived LW and WN ADMs are unavailable because of inadequate sampling. Therefore, fluxes for footprints over snow are estimated in the same manner as footprints with insufficient imager information (section 5c).

Precipitable water in Table 3 is the water vapor burden from the surface to the TOA. For scenes over water, the data source for precipitable water is the SSM/I dataset, if available. If SSM/I data are unavailable or the footprint is over land, ECMWF precipitable water is used. Under clear conditions, the vertical temperature change (ΔT_s) corresponds to the lapse rate in the first 300 hPa of the atmosphere above the surface. It is computed by subtracting the air temperature at the pressure level that is 300 hPa below the surface pressure (i.e., surface pressure minus 300 hPa) from the imager-based surface skin temperature. A separate ADM class is produced when there is an inversion in the boundary layer (ΔT_s < 0°C). When clouds are present in a CERES footprint, vertical temperature change (ΔT_c) refers to the difference in temperature between the surface and cloud; ΔT_c is computed by subtracting the imager-based effective (equivalent blackbody) cloud-layer temperature from the underlying skin temperature. If the imager-based surface skin temperature is unavailable (e.g., overcast conditions), a skin temperature from the ECMWF data assimilation model is used.

The mean infrared emissivity for a cloud layer is defined as the ratio of the difference between the observed and clear-sky VIRS 11-μm radiances to the difference between the cloud emission and clear-sky radiances. The clear-sky radiance is determined either from surrounding cloud-free observations when available, or from ECMWF surface skin temperature, surface emissivity maps, and ECMWF vertical profiles of temperature and humidity. The cloud emission radiance is defined as the blackbody radiance at the radiating temperature of the cloud. The mean infrared emissivity, cloud radiating temperature, and effective particle size are determined using an iterative procedure that minimizes the difference between calculated and observed VIRS radiances in the visible, solar-infrared, and infrared channels. The procedure relies on parameterizations described in Minnis et al. (1998) to relate cloud optical depth, particle size, and cloud infrared emissivity. Because scattering tends to block radiation from the warmer, lower portions of the cloud, the observed radiance can be less than the cloud emission radiance (i.e., the cloud appears colder than it really is). In these cases, the effective emissivity will be greater than 1. This occurs most often for optically thick clouds at large imager viewing zenith angles and for FOVs containing optically thick clouds that have an equivalent blackbody temperature that is within a few degrees of the clear-sky temperature (Minnis et al. 1998).

ADM sensitivity to scene type

Figure 14 provides examples of LW and WN ADMs for clear scenes over ocean, land, and desert as a function of ΔT_s for the 33d–66th-percentile interval of precipitable water. The ΔT_s intervals correspond to the 0–25th-, 25th–50th-, 50th–75th-, and >75th-percentile intervals (there was not sufficient sampling to construct ADMs for inversion conditions). The range of ΔT_s is much smaller over ocean than it is over land and desert because ΔT_s has a much narrower distribution over ocean. As ΔT_s increases, LW and WN anisotropy increases. For θ < 70°, the largest sensitivity in the LW anisotropic factors to ΔT_s occurs close to nadir, where it varies by ≈2%, which corresponds to a LW flux variation of ≈6 W m^–2. Sensitivity to ΔT_s is more pronounced for the WN channel because of its larger dependence on surface temperature.

LW and WN anisotropies change between daytime and nighttime conditions over clear desert regions (Figs. 15a–d). Because daytime surface temperatures are so much larger than those at night, the magnitude of ΔT_s in each percentile interval is greater during the daytime (Table 4). The daytime LW and WN anisotropy is consequently stronger than at night. Based on these results, ignoring daytime/nighttime differences in anisotropy would lead to a 5 W m^–2 bias in the day–night LW flux difference.

In broken cloud conditions, LW anisotropy shows a slight dependence on cloud infrared emissivity (Fig. 16, Table 5). Sensitivity to ε increases with cloud cover and cloud height, particularly over moist land. In general, LW anisotropy increases with decreasing ε. This trend is most pronounced for overcast conditions, as illustrated in Fig. 17 (see also Table 6), which shows nadir anisotropic factors for low and high overcast clouds. Clouds that occur in the wettest (largest precipitable water) regions with the largest ΔT_c show the strongest sensitivity to ε. Anisotropic factors range from 1.02 for thick high clouds to 1.158 for thin high clouds in the moist regions. This ≈13% difference in anisotropic factor corresponds to a difference in LW flux of 25 W m^–2 for these clouds.