Abstract

The Clouds and Earth’s Radiant Energy System (CERES) provides coincident global cloud and aerosol properties together with reflected solar, emitted terrestrial longwave, and infrared window radiative fluxes. These data are needed to improve the understanding and modeling of the interaction between clouds, aerosols, and radiation at the top of the atmosphere, surface, and within the atmosphere. This paper describes the approach used to estimate top-of-atmosphere (TOA) radiative fluxes from instantaneous CERES radiance measurements on the Terra satellite. A key component involves the development of empirical angular distribution models (ADMs) that account for the angular dependence of the earth’s radiation field at the TOA. The CERES Terra ADMs are developed using 24 months of CERES radiances, coincident cloud and aerosol retrievals from the Moderate Resolution Imaging Spectroradiometer (MODIS), and meteorological parameters from the Global Modeling and Assimilation Office (GMAO)’s Goddard Earth Observing System (GEOS) Data Assimilation System (DAS) V4.0.3 product. Scene information for the ADMs is from MODIS retrievals and GEOS DAS V4.0.3 properties over the ocean, land, desert, and snow for both clear and cloudy conditions. Because the CERES Terra ADMs are global, and far more CERES data are available on Terra than were available from CERES on the Tropical Rainfall Measuring Mission (TRMM), the methodology used to define CERES Terra ADMs is different in many respects from that used to develop CERES TRMM ADMs, particularly over snow/sea ice, under cloudy conditions, and for clear scenes over land and desert.

1. Introduction

One of the largest uncertainties in global climate models is the representation of how clouds and aerosols influence the earth’s radiation budget (ERB) at the surface, within the atmosphere, and at the top of the atmosphere. Because of the uncertainty in cloud–aerosol–radiation interactions, model predictions of climate change vary widely from one model to the next (Cess et al. 1990, 1996; Cubasch et al. 2001). To improve our understanding of cloud–aerosol–radiation interactions, and to identify key areas where climate models can be improved, global observations are needed. The central objective of the Clouds and the Earth’s Radiant Energy System (CERES) mission is to provide accurate global cloud, aerosol, and radiation data products to facilitate research addressing the role clouds and aerosols play in modulating the radiative energy flow within the earth–atmosphere system (Wielicki et al. 1996). The two CERES satellite instruments aboard the Terra spacecraft provide highly accurate shortwave (SW), longwave (LW), and infrared window (WN) radiance measurements and top-of-atmosphere (TOA) radiative flux estimates globally at a 20-km spatial resolution. These data, together with coincident cloud and aerosol properties inferred from the Moderate Resolution Imaging Spectroradiometer (MODIS), provide a consistent cloud–aerosol–radiation dataset for studying clouds and aerosols, and their influence on the ERB.

One of the challenges involved in producing ERB datasets from satellites is the need to convert the radiance measurements at a given sun–Earth–satellite configuration to outgoing reflected solar and emitted thermal TOA radiative fluxes. To estimate TOA fluxes from measured CERES radiances, one must account for the angular dependence in the radiance field, which is a strong function of the physical and optical characteristics of the scene (e.g., surface type, cloud fraction, cloud/aerosol optical depth, cloud phase, etc.), as well as the illumination angle. Because the CERES instrument can rotate in azimuth as it scans in elevation, it acquires data over a wide range of angles. Consequently, one can construct angular distribution models (ADMs) for radiance-to-flux conversion directly from the CERES measurements. Furthermore, because CERES and MODIS are on the same spacecraft, the ADMs can be derived as a function of MODIS-based scene-type parameters that have a strong influence on radiance anisotropy.

The first set of CERES ADMs were developed using 9 months of CERES and Visible Infrared Scanner (VIRS) data from the Tropical Rainfall Measuring Mission (TRMM) satellite between 38°S and 38°N from January to August 1998 and March 2000 (Loeb et al. 2003a). Because TRMM is in a 350-km circular precessing orbit with a 35° inclination angle, CERES TRMM sampled the full range of solar zenith angles over a region every 46 days. Unfortunately, the CERES TRMM instrument suffered a voltage-converter anomaly and acquired only 9 months of scientific data. In contrast, the CERES instruments on Terra have, thus far, acquired over 4 yr of global data with coarser spatial resolution (20 km versus 10 km for CERES TRMM) from a sun-synchronous orbit at an altitude of 705 km. Because of the differences in spatial resolution and geographical coverage between the CERES instruments on TRMM and Terra, direct application of the CERES TRMM ADMs to CERES Terra data is inappropriate, particularly at midlatitudes and in the polar regions.

The increased sampling that is available from CERES Terra provides a unique opportunity to develop a more comprehensive set of ADMs that are suitable for radiance-to-flux conversion with CERES Terra data and data from other broadband instruments with similar characteristics and orbital geometry. This paper is the first in a two-part series. Part I describes the development of new CERES Terra SW, LW, and WN ADMs from 2 yr of global data. Where appropriate, we compare the methodology used to produce CERES Terra ADMs with that used in Loeb et al. (2003a) to produce CERES TRMM ADMs. Part II will present extensive validation results in order to assess the accuracy of SW, LW, and WN TOA fluxes that are derived from the CERES Terra ADMs. TOA fluxes from the new Terra ADMs will also be compared with those from TRMM ADMs, as well as with fluxes based on algorithms developed during the Earth Radiation Budget Experiment (ERBE) (Smith et al. 1986; Suttles et al. 1992).

2. Observations

The Terra spacecraft, launched on 18 December 1999, carries two identical CERES instruments: Flight Model (FM)-1 and -2. Terra is in a descending sun-synchronous orbit with an equator-crossing time of 1030 LST. The CERES instrument is a scanning broadband radiometer that measures filtered radiances in the SW (wavelengths between 0.3 and 5 μm), total (TOT; wavelengths between 0.3 and 200 μm), and WN (wavelengths between 8 and 12 μm) regions. To correct for the imperfect spectral response of the instrument, the filtered radiances are converted to unfiltered reflected solar, unfiltered emitted terrestrial LW and WN radiances (Loeb et al. 2001). On Terra, CERES has a spatial resolution of approximately 20 km (equivalent diameter). One of the unique features of CERES is its ability to scan in either a fixed, rotating or programmable azimuth plane scan mode. Operationally, one CERES instrument is placed in a cross-track scan mode to optimize spatial sampling for time–space averaging (Young et al. 1998), while the second instrument is either in a rotating azimuth plane (RAP), an along-track, or a programmable azimuth plane (PAP) scan mode. In the RAP mode, the instrument scans in elevation as it rotates in azimuth thus acquiring radiance measurements from a wide range of viewing configurations. In PAP mode, CERES is programmed to collect measurements for a specific field campaign, for intercalibration with other instruments (e.g., CERES on TRMM, the Geostationary Earth Radiation Budget Instrument), or to augment sampling in specific viewing geometries (e.g., the principal plane). The nominal schedule is to operate the second CERES instrument in along-track mode every 15 days and in RAP mode the remainder of the time. In contrast, CERES TRMM was in RAP mode only every third day, and in along-track mode every 15 days. The increase in RAP sampling for CERES Terra, together with its relatively small range of solar zenith angle coverage relative to CERES TRMM (which sampled all solar zenith angles every 46 days), means that angular sampling at a particular solar zenith angle is increased by at least an order-of-magnitude for CERES Terra compared to CERES TRMM.

To construct ADMs for Terra, 24 months (March 2000–February 2002) of the CERES Terra Edition2A Single Scanner Footprint TOA/Surface Fluxes and Clouds (SSF) product (Geier et al. 2001) are used. The CERES SSF product combines CERES radiances and fluxes with scene information inferred from coincident MODIS measurements (Barnes et al. 1998) and meteorological fields based on 4D assimilation data. Cloud properties on the CERES Terra SSF product are inferred from MODIS pixel measurements using algorithms that are consistent with those used to produce cloud properties from VIRS (Kummerow et al. 1998) on the CERES TRMM SSF (Minnis et al. 2003). Aerosol properties are determined from two sources—(i) by applying the algorithm of Ignatov and Stowe (2002) and (ii) directly from the MOD04 aerosol product (Remer et al. 2005). For Terra Edition2A SSF, meteorological fields on the SSF are from the Global Modeling and Assimilation Office (GMAO)’s Goddard Earth Observing System (GEOS) Data Assimilation System (DAS) 4.0.3 product (DAO 1996). GMAO is running GEOS DAS V4.0.3 without any code modifications to produce a consistent analysis over the entire CERES data record.

As described in more detail in Loeb et al. (2003a), accurate spatial and temporal matching of imager-derived aerosol and cloud properties with CERES broadband radiation data are obtained by accounting for the CERES point spread function (PSF) (Smith 1994) when averaging imager-derived properties over the CERES footprint. 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 field of view (FOV), up to two nonoverlapping cloud layers are defined. A single footprint may contain any combination of a clear area and one or two distinct cloud layers (see Fig. 1 of Loeb et al. 2003a). To reduce the processing time needed to generate the CERES SSF product, the CERES team has decided to process only every fourth MODIS pixel from every second scan line. This introduces a random noise in the PSF-weighted average imager reflectance and brightness temperature of approximately 1.5% and 0.2%, respectively, over CERES FOVs (W. F. Miller 2003, personal communication).

3. CERES Terra ADM development

TOA flux is the radiant energy emitted or scattered by the earth–atmosphere per unit area. Flux is related to radiance (I) as follows:

 
formula

where θo is the solar zenith angle, θ is the observer viewing zenith angle, and ϕ is the relative azimuth angle defining the azimuth angle position of the observer relative to the solar plane. An ADM is a set of anisotropic factors (R) for determining the TOA flux from an observed radiance as follows:

 
formula

Because CERES measures the upwelling radiation from a scene at any given time from one or more directions, F (or R) cannot be measured instantaneously. Instead, R is obtained from a set of predetermined empirical ADMs that are defined for several scene types with distinct anisotropic characteristics. For CERES TRMM, the ADMs were constructed by the sorting-into-angular-bins (SABs) method (Suttles et al. 1992; Loeb et al. 2003a). In the SAB method, a large ensemble of radiance measurements are first sorted into discrete angular bins and parameters that define an ADM scene type, and ADM anisotropic factors for a given scene type (j) are given by

 
formula

where Ij is the average radiance (corrected for Earth–sun distance in the SW) in an angular bin (θoi, θk, ϕl), and Fj is the upwelling flux in a solar zenith angle bin θoi, which is determined by directly integrating Ij over all angles (Loeb et al. 2003a). The set of angles (θoi, θk, ϕl). corresponds to the midpoint of a discrete angular bin defined by [θoi ± (Δθo/2), θk ± (Δθ/2), ϕl ± (Δϕ/2)], where Δθo, Δθ, and Δϕ represent the angular bin resolution. To evaluate Fj from satellite measurements, Loeb et al. (2002) showed that the reference level for the satellite viewing geometry must be defined at least 100 km above the earth’s surface in order to account for radiation contributions escaping the planet along slant paths above the earth’s tangent point. Loeb et al. (2002) also argue that the optimal reference level for defining instantaneous 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.

For CERES Terra these definitions are retained. The SAB method is used to develop ADMs for some, but not all, ADM scene types. As described in more detail in sections 4 and 5, where possible, we have developed ADMs that are continuous functions of imager-based retrievals, using analytical functions to represent the CERES radiance dependence on scene type. As in Loeb et al. (2003a), SW ADMs are defined explicitly as a function of three angles (θoi, θk, ϕl), while LW ADMs are assumed to be a function of only viewing zenith angle.

4. SW ADMs

a. Ocean

1) Clear

When the MODIS pixel-level cloud coverage within a CERES FOV is ≤0.1%, the CERES FOV is defined as being clear. Following the approach used in Loeb et al. (2003a), instantaneous TOA fluxes are determined using a combination of empirical and theoretical ADMs as follows:

 
formula

where R(wk, θo, θ, ϕ) is determined from wind speed–dependent empirical ADMs that are derived from CERES data, and Rth(wk, I) and Rth(wk, I) are theoretical radiative transfer model anisotropic factors evaluated at the measured CERES radiance I(θo, θ, ϕ) and mean CERES radiance I(wk, θo, θ, ϕ) in a given ADM angular bin, respectively. For Terra, the angular bin resolution of the clear ocean SW ADMs has been sharpened to 2° (CERES TRMM used 10° for θo and θ, and 20° for ϕ), and the wind speed resolution has been increased to 2 m s−1 for winds ranging from 0 to 12 m s−1 (the nominal wind speed intervals for CERES TRMM were <3.5, 3.5–5.5, 5.5–7.5, and >7.5 m s−1). Wind speeds correspond to the 10-m level based on Special Sensor Microwave Imager (SSM/I) retrievals (Goodberlet et al. 1990) that have been ingested into the GEOS-4 DAS analysis. To determine Rth(wj, I) and Rth(wj, I), CERES radiances I(θo, θ, ϕ; hsfc) and I(θo, θ,ϕ; hsfc) are compared with lookup tables of theoretical SW radiances stratified by aerosol optical depth. Here, Rth(wj, I) and Rth(wj, I) correspond to the aerosol optical depth for which the theoretical radiances match the CERES radiances. The radiative transfer calculations are from the rstar5b radiative transfer model that is based on Nakajima and Tanaka (1986, 1988). Maritime aerosols from Hess et al. (1998) evaluated at 26 optical depths are used to produce the lookup tables. The ocean surface in the calculations accounts for the bidirectional reflectance of the ocean at wind speeds that correspond to the midpoints of the CERES ADM wind speed intervals.

2) Clouds

The CERES TRMM ADMs were developed for discrete scene types defined by cloud phase (two categories), cloud fraction (12 intervals), and cloud optical depth (14 intervals). Because of the close spatial and temporal coincidence between CERES radiances and imager-derived parameters over a CERES FOV, an alternate approach is to construct continuous ADMs using analytical functions that relate CERES radiances and imager parameters. For clouds, the magnitude of CERES radiances in an angular bin is most sensitive to cloud fraction and cloud optical depth. To illustrate, Figs. 1a and 1b show instantaneous CERES SW radiances in the angular bin, defined by θo = 34°–36°; θ = 50°–52°; ϕ = 6°–8° against cloud fraction ( f) (Fig. 1a) and the mean logarithm of cloud optical depth (lnτ̃) (Fig. 1b) for liquid water cloud layers. Here, τ̃ is defined by

 
formula

and τi is the retrieved cloud optical depth of the ith pixel within the CERES FOV. Four months (November–December 2000, April–May 2001) of SSF data are considered in Fig. 1. Equation (5) follows from Cahalan et al. (1994), who showed that for overcast conditions albedo is approximately linear in when either the variability in the cloud optical depth field is small or the curvature in the albedo relation is small. When both overcast and broken cloud fields are considered, the SW radiance dependence on f and lnτ̃ shows a lot of scatter (Figs. 1a–b). To improve the relationship, we seek to combine f and lnτ̃ into a single parameter. Figures 1c and 1d show results for two candidates: (i) lnτ̃ weighted by cloud fraction over a CERES FOV, and (ii) the logarithm of τ̃ weighted by cloud fraction, respectively. Mathematically, these are expressed as

 
formula
 
formula

In Fig. 1c, a third-order polynomial fit (solid black line) is applied to data points, while Fig. 1d applies a five-parameter sigmoidal fit, defined by

 
formula

where xo, Io, a, b, c are coefficients of the fit, and x = ln( fτ̃). The relative error in the fits for different intervals of cloud fraction is shown in Fig. 1e. The sigmoidal fit relative error remains less than 1% in every cloud fraction interval, while it reaches −3% at intermediate cloud fractions using the polynomial fit. The relative root-mean-square (rms) error for the two fits is similar, approximately 8.7% for the polynomial fit and 8.6% for the sigmoidal fit. Similar results are obtained when other angular bins are considered or when separate fits are derived for mixed-phase and ice clouds (not shown). In general, the rms error in predicting instantaneous SW radiances using the sigmoidal fit is between 5% and 10%. The close relationship between SW radiance and ln( fτ̃) occurs in spite of the rather large range of cloud properties associated with a given ln( fτ̃) range. This is illustrated in Fig. 1fwhich shows ln( fτ̃) against f and τ̃ [in ln( fτ̃) increments of 1]. At intermediate ln( fτ̃) (e.g., from 0 to 1), f varies by as much as 0.8 (e.g., from 0.2 to 1.0) and τ̃ varies by 4 (e.g., from 1 to 5). In contrast, the corresponding SW radiance variability is only 8.5%.

Fig. 1.

Instantaneous CERES SW radiances for liquid water cloud layers in the angular bin defined by θo = 34°–36°, θ = 50°–52°, and ϕ = 6°–8° against (a) cloud fraction ( f); (b) mean logarithm of cloud optical depth (lnτ̃); (c) f lnτ̃; and (d) ln( fτ̃). Solid lines in (c) and (d) correspond to a third-order polynomial fit and a five-parameter sigmoid fit, respectively. (e) Relative error in the fits for different intervals of cloud fraction and (f) ln( fτ̃) against f and τ̃.

Fig. 1.

Instantaneous CERES SW radiances for liquid water cloud layers in the angular bin defined by θo = 34°–36°, θ = 50°–52°, and ϕ = 6°–8° against (a) cloud fraction ( f); (b) mean logarithm of cloud optical depth (lnτ̃); (c) f lnτ̃; and (d) ln( fτ̃). Solid lines in (c) and (d) correspond to a third-order polynomial fit and a five-parameter sigmoid fit, respectively. (e) Relative error in the fits for different intervals of cloud fraction and (f) ln( fτ̃) against f and τ̃.

CERES Terra ADMs are determined from sigmoidal fits between SW radiance and ln( fτ̃) in 2° angular bins (i.e., 2° resolution for θo, θ, and ϕ) as a function of cloud phase. Cloud phase is represented by an effective cloud phase (ECP) index (Loeb et al. 2003a), which is a PSF-weighted average of cloud phase derived from imager pixel data (1 = liquid water, and 2 = ice). For CERES TRMM, “liquid clouds” were defined as footprints with ECP < 1.5, and “ice clouds” were defined as footprints with ECP ≥ 1.5. For CERES Terra, ADMs are defined for three categories of cloud phase: liquid water (1.00 < ECP < 1.01), mixed phase (1.01 ≤ ECP ≤ 1.75), and ice (1.75 < ECP ≤ 2.00).

In angular bins where sunglint is strong (i.e., within 20° of the specular reflection direction), sigmoidal fits are defined only for thick clouds [ln( fτ̃) ≥ 1.4 or fτ̃ ≥ 4]. For thin clouds in sunglint, SW radiances are averaged in four discrete intervals of ln( fτ̃). To determine sigmoidal fits using all of the available CERES Terra measurements (i.e., 24 months), CERES SW radiances are first averaged in 750 intervals of ln( fτ̃) between −10 and 5. The TOA flux in each ln( fτ̃) interval is obtained by integrating SW radiances inferred from the fits in all upwelling directions. Figure 2 shows an example of TOA flux against ln( fτ̃) for liquid water clouds at θo = 44°–46°. Anisotropic factors at a given value of ln( fτ̃) are determined from an expression similar to Eq. (3), using sigmoidal fits to infer radiances and lookup tables of SW TOA flux as a function of solar zenith angle and ln( fτ̃).

Fig. 2.

TOA flux against ln( fτ̃) for liquid water clouds at θo = 44°–46°.

Fig. 2.

TOA flux against ln( fτ̃) for liquid water clouds at θo = 44°–46°.

Figures 3a and 3b show CERES SW anisotropic factors in the principal plane for liquid water (Fig. 3a) and ice clouds (Fig. 3b) at ln( fτ̃) = 2.01 (or fτ̃ = 7.5) for three solar zenith angle intervals based on 24 months of CERES Terra measurements. In each solar zenith angle interval, the liquid water clouds show well-defined peaks in anisotropy for θ = −30° to −60° and close to nadir due to the cloud glory and rainbow features, respectively, while peaks in anisotropy occur for ice clouds between θ = 30° and 60° in the specular reflection direction. Chepfer et al. (1999) also observed these features in multiangle Polarization and Directionality of Earth Reflectances (POLDER) measurements and showed theoretically that these are likely due to horizontally oriented ice crystals. Such pronounced microphysical features were not present in ERBE and CERES TRMM ADMs because the angular bins used to define those ADMs were too coarse.

Fig. 3.

CERES SW anisotropic factors in the principal plane for (a) liquid water and (b) ice clouds at ln( fτ̃) = 2.01 (or fτ̃ = 7.5) for three solar zenith angle intervals based on 24 months of CERES Terra measurements (negative θ corresponds to ϕ = 178°–180°; positive θ corresponds to ϕ = 0°–2°).

Fig. 3.

CERES SW anisotropic factors in the principal plane for (a) liquid water and (b) ice clouds at ln( fτ̃) = 2.01 (or fτ̃ = 7.5) for three solar zenith angle intervals based on 24 months of CERES Terra measurements (negative θ corresponds to ϕ = 178°–180°; positive θ corresponds to ϕ = 0°–2°).

b. Land and desert ADMs

1) Clear

Over clear land and desert, the CERES TRMM ADMs were defined by grouping the International Geosphere Biosphere Program (IGBP) global land cover types (Loveland and Belward 1997) over the Tropics into four categories: low-to-moderate tree/shrub coverage, moderate-to-high tree/shrub coverage, dark desert, and bright desert. These categories were assumed to apply over extensive geographical areas and remain invariant throughout the year. On a global scale, this classification is inadequate because it does not account for vegetation types outside of the Tropics, many of which exhibit strong seasonal variations. To improve the spatial resolution of the clear land and desert ADMs and account for changes in surface type with season, the Terra ADMs over clear land and desert are defined for 1° latitude × 1° longitude equal area regions, with a temporal resolution of 1 month. To generate ADMs at these scales, all snow-free, clear-sky CERES Terra SW radiances from the available 24 months of SSF data are first sorted by calendar month and 1° latitude × 1° longitude equal area region. Each FOV radiance is converted to a reflectance as follows:

 
formula

where μo = cosθo, Eo is the TOA solar irradiance (=1365 W m−2) and (1 + ɛse) is the Earth–sun distance in astronomical units (AUs). A TOA normalized vegetation difference index (NDVI) for each CERES FOV is determined from PSF-weighted mean MODIS 0.63- (I0.63) and 0.86-μm (I0.86) radiances as follows:

 
formula

The TOA NDVI is used to separate subregions within a 1° latitude × 1° longitude region that have different vegetation characteristics. Subregions with TOA NDVI differing by 0.1 or more are treated separately. Next, if angular sampling within a region is sufficient, an eight-parameter nonparametric fit from Ahmad and Deering [1992see their Eq. (37)] is applied to the CERES SW reflectances to represent the angular dependence in the reflectance field. The Bidirectional Reflectance Distribution Function (BRDF) used in the fit accounts for multiple scattering based on Chandrasekhar’s (1950) radiative transfer solution for a semi-infinite medium, and the so-called “hot spot” is modeled using an empirical term (Hapke 1986). Separate fits are derived for every 0.2 increment in μo, provided that at least three CERES FOVs are available in the following geometries: (i) θ ≤ 20°, (ii) θ ≥ 40° and ϕ ≤ 30°, (iii) θ ≥ 40° and 60° ≤ ϕ ≤ 120°, and (iv) θ ≥ 40° and ϕ ≥ 150°. If this condition is not satisfied, then CERES FOVs from neighboring regions with the same IGBP type, NDVI and μo intervals are used to supplement the angular sampling. Only FOVs from neighboring regions within ±15° latitude × ±15° longitude are considered. If the viewing angle sampling criterion is still not satisfied, then a fit is not performed, and fluxes are determined using the CERES TRMM ADMs.

Figures 4a and 4b show the regional relative rms error when the BRDF fits are applied to RAP and cross-track CERES data from December 2000 through February 2001 (Fig. 4a), and from June 2000 through August 2000 (Fig. 4b). Histograms of relative error and relative rms error are provided in Figs. 5a–b. Overall, the relative rms error in reflectance from the BRDF fit is between 6% and 7% for the two seasons. Relative errors tend to be larger over mountainous regions (e.g., Rockies, Andes, Tibetan Plateau) and smaller over the broadleaf forest regions of South America and over the central United States in summer.

Fig. 4.

Regional relative rms error in BRDF fits for RAP and cross-track CERES data from (a) Dec 2000 through Feb 2001 and (b) Jun 2000 through Aug 2000.

Fig. 4.

Regional relative rms error in BRDF fits for RAP and cross-track CERES data from (a) Dec 2000 through Feb 2001 and (b) Jun 2000 through Aug 2000.

Fig. 5.

Histograms of (a) 1° regional relative bias and (b) 1° regional relative rms error in BRDF fits for RAP and cross-track CERES data from Dec 2000 through Feb 2001, and Jun 2000 through Aug 2000.

Fig. 5.

Histograms of (a) 1° regional relative bias and (b) 1° regional relative rms error in BRDF fits for RAP and cross-track CERES data from Dec 2000 through Feb 2001, and Jun 2000 through Aug 2000.

To construct an ADM from the BRDF fits, albedos at several solar zenith angles in the interval of μo, in which the BRDF fit was derived, are first computed by directly integrating the BRDFs over θ and ϕ. Next, a fit based on Rahman et al. (1993) is used to represent the albedo dependence on solar zenith angle in each μo interval. The instantaneous anisotropic factor at a given location is inferred from the ratio of reflectance to albedo, both of which are evaluated from the above fits at the FOV viewing geometry.

2) Clouds

The anisotropy of clouds over land and desert depends strongly upon cloud phase, cloud fraction, cloud optical thickness, and the underlying surface type (especially in thin or broken cloud conditions). CERES TRMM ADMs for clouds over land and desert were defined for discrete classes of cloud phase, cloud fraction, and cloud optical depth for the four surface types that were used to define clear land CERES TRMM ADMs. For CERES Terra, we use a similar approach to that outlined over the ocean [section 4b(2)], with an additional correction to account for surface reflection. Following Arking and Childs (1985), an observed radiance (I) is modeled as follows:

 
formula

where the first term corresponds to reflection from the cloud-free area, the second term represents reflection from the cloud, and the third and fourth terms correspond to scattering by the surface and atmosphere transmitted through the cloud, respectively; ρclr is the clear-sky bidirectional reflectance, μ = cosθ, I cld is the radiance from the cloud layer, τ is the cloud optical depth, clr and cld are the clear-sky and cloud spherical albedos (Thomas and Stamnes 1999), respectively, and tcld is the diffuse transmittance of the cloud. Respectively, ρclr and clr are inferred from 1° BRDF fits; cld and tcld are determined from broadband radiative transfer model calculations using a highly modified version of the model described in Fu and Liou (1993). In the radiative transfer model calculations, liquid water clouds are assumed to have an effective droplet radius of 10 μm, and ice clouds are assumed to have an effective particle diameter of 60 μm. Lookup tables of cld and tcld are generated at 17 solar zenith angles and 19 cloud optical depths.

Because I is known from the observations, Eq. (11) can be used to estimate the contribution from the cloudy area ( fI cld). Figure 6a shows an example of fI cld against ln( fτ̃) for the angular bins θo = 40°–45°, θ = 0°–5°, and ϕ = 0°–5°. To generate this curve, all CERES measurements from 24 months for all land and desert types were included and CERES radiances were averaged in 375 intervals of ln( fτ̃). The relative rms error in the sigmoid fit (solid line) is approximately 7%, comparable to what was obtained for clouds over the ocean. Sigmoidal fits were also determined in all other angular bins where CERES Terra observations occur. An angular resolution of 5° in θo, θ, and ϕ is used for land and desert. Separate fits were derived for liquid water, and mixed and ice clouds, defined in section 4a(2). Using fI cld, predicted from the sigmoidal fits together with the 1° latitude × 1° longitude monthly clear land BRDF fits described in section 4b(1), the approximation in Eq. (11) is used to construct ADMs for clouds over land and desert that account for regional and seasonal changes in surface properties. The ADM flux is inferred by integrating Eq. (11):

 
formula

where αclr is the plane albedo and tcld is the spherical transmittance (ratio of transmitted flux to incident flux for an isotropic source). To reduce computation time, we assume ρclr(μo, μ, ϕ) ≈ αclr(μo) in the third term of Eq. (12), thereby avoiding explicit double integration over ρclr(μo, μ, ϕ) eτ/μ for every CERES FOV.

Fig. 6.

(a) Cloud SW radiance ( f Icld) against ln( fτ̃) for clouds over land and desert in angular bins θo = 40°–45°, θ = 0°–5°, and ϕ = 0°–5°. Solid line is a five-parameter sigmoid fit to the data. (b) Sample ADMs for clear and cloudy conditions over a cropland/natural vegetation mosaic surface (latitude = 36.52°N, longitude = 128.72°E) for θo = 59.24° on 2 Dec 2000. The cloud is composed of liquid water, covers 74% of the CERES FOV, and has a cloud optical depth of 5.2.

Fig. 6.

(a) Cloud SW radiance ( f Icld) against ln( fτ̃) for clouds over land and desert in angular bins θo = 40°–45°, θ = 0°–5°, and ϕ = 0°–5°. Solid line is a five-parameter sigmoid fit to the data. (b) Sample ADMs for clear and cloudy conditions over a cropland/natural vegetation mosaic surface (latitude = 36.52°N, longitude = 128.72°E) for θo = 59.24° on 2 Dec 2000. The cloud is composed of liquid water, covers 74% of the CERES FOV, and has a cloud optical depth of 5.2.

An anisotropic factor for an arbitrary FOV is determined from radiance and flux estimates using Eqs. (11) and (12) with the appropriate clear-sky 1° BRDF fits and sigmoidal curve. Figure 6b provides sample ADMs for clear and cloudy conditions over a cropland/natural vegetation mosaic surface (latitude = 36.52°N, longitude = 128.72°E) for θo = 59.24° on 2 December 2000. The cloud is composed of liquid water, covers 74% of the CERES FOV, and has a cloud optical depth of 5.2. The clear-sky case shows a markedly stronger backscatter contribution compared to the cloud case, which scatters more radiation into the forward direction owing to its scattering phase function characteristics.

c. Snow and sea ice

One of the major differences in angular distribution model development for Terra compared with TRMM is the availability of CERES RAP data over polar regions. Because the TRMM orbit is restricted to tropical latitudes, there were not enough data to develop empirical snow ADMs for CERES TRMM. As a result, Loeb et al. (2003a) used theoretical ADMs to infer TOA fluxes over snow in tropical regions. Because Terra is a sun-synchronous polar-orbiting satellite, CERES instruments on Terra measure radiances in polar regions from various scene types and a wide range of viewing geometries. This allows the development of empirical ADMs to estimate radiative fluxes from snow and sea ice.

For convenience, snow/ice surfaces are divided into three groups: permanent snow, fresh snow, and sea ice. Most permanent snow scenes occur over Antarctica and Greenland, whereas fresh snow and sea ice occur over land and water, respectively. Because anisotropy also varies with surface brightness (Loeb et al. 2003a), each of the three surface types are further stratified into “bright” and “dark” subclasses. A CERES FOV is determined to be bright or dark by comparing its geographical location with a predetermined monthly regional snow map that classifies all 1° × 1° regions with snow/sea ice as either bright or dark (Kato and Loeb 2005). The snow maps are constructed as follows. (i) Using all available cloud-free CERES FOVs with snow/sea ice, mean MODIS 0.63-μm near-nadir (for θ < 25°) reflectances are determined as a function of snow type and solar zenith angle; (ii) every CERES FOV whose MODIS 0.63-μm near-nadir reflectance lies below (above) the corresponding mean reflectance is assigned a value of −1 (+1); (iii) if the sum of all CERES FOV classifications in a 1° × 1° region from 1 month of data is negative (positive), the region is classified as dark (bright). In this manner, 12 snow maps representing each calendar month are produced.

To account for the effects of partial coverage by fresh snow or sea ice within a CERES FOV on anisotropy, bright and dark fresh snow and sea ice ADMs are further stratified into six intervals of fresh snow or sea ice percent coverage. When clouds are present, ADMs are further stratified by cloud fraction and cloud optical thickness. Table 1 shows how the snow and sea ice ADMs are defined for each surface type. The total number of ADMs is 10 for permanent snow, 25 for fresh snow, and 25 for sea ice.

Table 1.

SW ADM scene-type definitions for permanent snow, fresh snow, and sea ice.

SW ADM scene-type definitions for permanent snow, fresh snow, and sea ice.
SW ADM scene-type definitions for permanent snow, fresh snow, and sea ice.

Following Loeb et al. (2003a), radiances measured by CERES instruments are sorted into angular bins and averaged. Angular bin sizes are 2° for the solar zenith angle, and 5° for both viewing zenith and relative azimuth angles over permanent snow. For fresh snow and sea ice, angle bin sizes are 5° for all three angles. Radiances in undersampled angular bins are inferred using the approach outlined in Loeb et al. (2003a).

Figures 7a–c show SW anisotropic factors for permanent snow (Fig. 7a), fresh snow (Fig. 7b), and sea ice (Fig. 7c) as a function of θ for ϕ = 0°–10° and ϕ = 170°–180°. When clouds are present over snow/sea ice, SW anisotropic factors show a greater dependence on viewing zenith angle than cloud-free scenes, especially in the forward scattering direction. Anisotropic factors for clear bright and dark surfaces are remarkably similar over permanent snow, while the brighter surfaces tend to be slightly more isotropic than dark scenes over fresh snow and sea ice.

Fig. 7.

SW anisotropic factors against CERES viewing zenith angle near the principal plane for (a) permanent snow for θo = 62°–64°; (b) fresh snow for θo = 60°–65°; and (c) sea ice for θo = 60°–65°. “Clr” corresponds to clear; “Ovc” corresponds to overcast.

Fig. 7.

SW anisotropic factors against CERES viewing zenith angle near the principal plane for (a) permanent snow for θo = 62°–64°; (b) fresh snow for θo = 60°–65°; and (c) sea ice for θo = 60°–65°. “Clr” corresponds to clear; “Ovc” corresponds to overcast.

d. Mixed-scene fields of view

Shortwave anisotropic factors for CERES FOVs that lie over water–land–snow boundaries are determined by accounting for the fractional coverage by each surface type as follows:

 
formula

where fW, fL, and fS correspond to the fractional coverage over a CERES FOV by water, land, and snow, respectively; and IX and FX (X = W, L, S) are the mean radiance and TOA flux used to define ADMs for FOVs with 100% coverage by water, land, or snow.

e. Sunglint conditions

When a CERES FOV is over water and the satellite viewing geometry is near the specular reflection direction, the radiance-to-flux conversion is less reliable owing to the large variability in ocean reflectance at those angles. To determine whether or not a footprint is too close to the specular reflection direction to provide a reliable flux retrieval, the following expression is evaluated:

 
formula

where fice and fcld correspond to the fraction of the CERES FOV covered by sea ice and cloud, respectively, and σRclr is the standard deviation of clear ocean anisotropic factors in angular bins adjacent to the observation angle. If an observation falls in an angular bin for which σR ≥ 0.05, a radiance-to-flux conversion is not performed. Instead, a mean flux value, corresponding to the ADM scene type over the FOV, is used. ADM flux values are determined when the ADMs are constructed by direct integration of the radiances for the corresponding scene type.

5. LW and WN ADMs

ADMs for LW and WN scenes are defined in terms of several surface and meteorological properties that influence radiance anisotropy over the ocean, land, and desert. In addition, because the cloud retrieval algorithm uses different approaches during the daytime and nighttime owing to the lack of visible imager information at night, separate LW and WN ADMs are developed for daytime and nighttime conditions.

a. Clear ocean, land, and desert

To account for the increased variability in surface properties encountered by Terra compared to TRMM, the number of surface types used to define land and desert ADMs has been increased from two for TRMM to six for Terra. Table 2 provides the IGBP surface types corresponding to each of the six land categories. These classes were determined by analyzing the spatial distribution of surface emissivity (Wilber et al. 1999) over the different IGBP types.

Table 2.

Surface-type definitions for clear-sky LW and WN ADMs over the ocean, land, and desert.

Surface-type definitions for clear-sky LW and WN ADMs over the ocean, land, and desert.
Surface-type definitions for clear-sky LW and WN ADMs over the ocean, land, and desert.

In addition to surface type, the scene types are stratified into discrete intervals of precipitable water (w), vertical temperature change (ΔT), and imager-based surface skin temperature (Ts) (Table 3). Over water, w is obtained from SSM/I retrievals; over land and desert, w is obtained from meteorological values (DAO 1996). Here, ΔT is defined as the lapse rate in the first 300 hPa of the atmosphere above the surface. It is computed by subtracting the DAO (1996) air temperature at the pressure level that is 300 hPa below the surface pressure (i.e., surface pressure minus 300 hPa) from Ts; Ts is estimated from the clear-sky 11-μm radiance using a narrowband radiative transfer algorithm that uses temperature and humidity profile inputs from the GEOS DAS V4.0.3 (Minnis et al. 2003).

Table 3.

Precipitable water (w), lapse rate (ΔT), and surface skin temperature (Ts) intervals used to determine LW and WN ADMs under clear-sky conditions over the ocean, land, and desert.

Precipitable water (w), lapse rate (ΔT), and surface skin temperature (Ts) intervals used to determine LW and WN ADMs under clear-sky conditions over the ocean, land, and desert.
Precipitable water (w), lapse rate (ΔT), and surface skin temperature (Ts) intervals used to determine LW and WN ADMs under clear-sky conditions over the ocean, land, and desert.

Longwave and WN ADMs are defined as a function of viewing zenith angle using a 2° angular bin resolution. Consequently, variations in anisotropy with solar zenith angle and relative azimuth angle are not accounted for. While this approximation is reasonable for the ocean and for all surface types at night, it breaks down during daytime for land areas with highly variable topography (Minnis and Khaiyer 2000; Minnis et al. 2004).

Figures 8a–d provide examples of LW ADMs for different surface types as a function of surface skin temperature for w < 1 and ΔT between 15 and 30 K. For all surface types, LW anisotropy increases as surface skin temperature increases. Because the WN channel is more sensitive to surface skin temperature, WN anisotropy (not shown) is found to be significantly more pronounced than LW anisotropy. While LW and WN anisotropy also increases with ΔT, the sensitivity is less pronounced than it is to Ts.

Fig. 8.

LW anisotropic factors for clear scenes with w = 0.0–1.0 cm and ΔT = 15°–30°C; (a) ocean, (b) forests, (c) savannas, (d) cropland/grassland, (e) dark deserts, and (f) bright deserts.

Fig. 8.

LW anisotropic factors for clear scenes with w = 0.0–1.0 cm and ΔT = 15°–30°C; (a) ocean, (b) forests, (c) savannas, (d) cropland/grassland, (e) dark deserts, and (f) bright deserts.

b. Clouds over the ocean, land, and desert

Under cloudy conditions, LW and WN anisotropy depends on several parameters, including surface type, w, Ts, surface–cloud temperature difference (ΔTsc), cloud fraction, and cloud infrared emissivity (ɛc). To characterize scenes in terms of these parameters, we define a “pseudoradiance” parameter ψ as follows:

 
formula

where fj is the cloud fraction of the jth cloud layer within a CERES FOV ( f = f1 + f2), Tcj is the corresponding layer cloud-top temperature, ɛs is the surface infrared emissivity, and B(T) = π−1σT 4, where σ is the Stefan–Boltzmann constant (=5.6696 × 10−8 W m−2 sr−1 deg−4). Here, ɛcj is determined from

 
formula

where τaj is the infrared absorption cloud optical depth of the jth layer derived using the approach outlined in Minnis et al. (1998) from visible cloud optical depth and particle effective radius retrievals available on the SSF product. For a given surface type, and fixed intervals of w, f, Ts, and ΔTsc (Table 4), LW (and WN) radiances show a simple monotonic dependence on ψ. As an example, Fig. 9a shows LW radiances against ψ for three different viewing bins over the ocean for w > 5 cm, Ts=300–305 K, ΔTsc >85 K, and f=1.0. To produce Fig. 9a, the observed LW radiances were averaged in 250 ψ bins of a width of 1 W m−2 sr−1. Note that because w, f, Ts, and ΔTsc are held fixed when Eq. (15) is applied, the main source of variation in ψ is from ɛc For an arbitrary CERES FOV, R is determined by first evaluating the radiance at ψ from Eq. (15) in each viewing zenith angle bin. The relationship between radiance and ψ in a given angular bin is derived from predetermined third-order polynomial fits in each θ bin for the intervals of w, f, Ts, and ΔTsc, shown in Table 4. The radiances are then integrated over viewing zenith angle to produce an ADM flux and R is obtained directly from Eq. (3). Figure 9b shows the viewing zenith angle dependence of R at ψ = 33.8 W m−2 sr−1 and ψ = 95.8 W m−2 sr−1, corresponding to the conditions in Fig. 9a. At ψ = 33.8 W m−2 sr−1, the cloud is thick (ɛc ≈ 1) and the viewing zenith angle dependence of R is weak. This is expected since the cloud top is located in the upper troposphere where there is less attenuation above the cloud to cause appreciable limb darkening. At ψ = 95.8 W m−2 sr−1, the clouds are much thinner (ɛc ≈ 0.4–0.5) and the LW anisotropy is more pronounced because the contribution from the warm ocean surface transmitted through the cloud is attenuated more rapidly with viewing zenith angle.

Table 4.

Surface type, precipitable water (w), cloud fraction ( f), surface–cloud temperature difference (ΔTsc), and surface skin temperature (Ts) intervals used to determine LW and WN ADMs under cloudy conditions over the ocean, land, and desert.

Surface type, precipitable water (w), cloud fraction ( f), surface–cloud temperature difference (ΔTsc), and surface skin temperature (Ts) intervals used to determine LW and WN ADMs under cloudy conditions over the ocean, land, and desert.
Surface type, precipitable water (w), cloud fraction ( f), surface–cloud temperature difference (ΔTsc), and surface skin temperature (Ts) intervals used to determine LW and WN ADMs under cloudy conditions over the ocean, land, and desert.
Fig. 9.

(a) LW radiance against ψ for θ = 0°–2°, θ = 30°–32°, and θ = 68°–70° over the ocean for w > 5 cm, Ts = 300–305 K, ΔTsc > 85 K, and f = 1.0. Solid line corresponds to a third-order polynomial fit to the data. (b) Anisotropic factor (R) against CERES viewing zenith angle for ψ = 33.8 W m−2 sr−1 and ψ = 95.8 W m−2 sr−1.

Fig. 9.

(a) LW radiance against ψ for θ = 0°–2°, θ = 30°–32°, and θ = 68°–70° over the ocean for w > 5 cm, Ts = 300–305 K, ΔTsc > 85 K, and f = 1.0. Solid line corresponds to a third-order polynomial fit to the data. (b) Anisotropic factor (R) against CERES viewing zenith angle for ψ = 33.8 W m−2 sr−1 and ψ = 95.8 W m−2 sr−1.

c. Snow

Longwave and WN ADMs over permanent snow, fresh snow, and sea ice are defined with an angular resolution of 2° in viewing zenith angle for 24 discrete scene classes by clear fraction, surface skin temperature, and surface–cloud top temperature difference (Table 5). Figures 10a–c show daytime LW ADMs for the three surface types. As expected, LW ADMs for clear scenes with Ts > 250 K are more anisotropic than those with Ts ≤ 250 K. Under cloudy conditions, larger anisotropy occurs when Ts > 250 K and ΔTsc > 20 K. Clouds in this scene type are not completely opaque close to nadir, so that the difference in the effective temperature at nadir and the oblique viewing angle is large. For θ > 84°, the radiances show more variability because of reduced sampling and because part of the CERES FOV lies beyond the earth’s horizon (no scene information is available from the imager over that part of the FOV). The uncertainty in TOA flux due to radiance uncertainties at θ > 84° is <0.3 W m−2.

Table 5.

Clear fraction ( fclr), surface skin temperature (Ts), and surface–cloud temperature difference (ΔTsc) intervals used to determine LW and WN ADMs over permanent snow (PS), fresh snow, and sea ice.

Clear fraction ( fclr), surface skin temperature (Ts), and surface–cloud temperature difference (ΔTsc) intervals used to determine LW and WN ADMs over permanent snow (PS), fresh snow, and sea ice.
Clear fraction ( fclr), surface skin temperature (Ts), and surface–cloud temperature difference (ΔTsc) intervals used to determine LW and WN ADMs over permanent snow (PS), fresh snow, and sea ice.
Fig. 10.

Daytime LW anisotropic factors against CERES viewing zenith angle for (a) permanent snow, (b) fresh snow, and (c) sea ice. “Clr” corresponds to clear; “Ovc” corresponds to overcast; Tsfc refers to surface skin temperature; Tcld corresponds to cloud-top temperature.

Fig. 10.

Daytime LW anisotropic factors against CERES viewing zenith angle for (a) permanent snow, (b) fresh snow, and (c) sea ice. “Clr” corresponds to clear; “Ovc” corresponds to overcast; Tsfc refers to surface skin temperature; Tcld corresponds to cloud-top temperature.

6. Footprints with insufficient imager information

In circumstances where there is insufficient imager coverage or scene information for a CERES FOV due to missing MODIS data and/or missing cloud property retrievals, anisotropic factors are determined from the CERES radiances directly using a feed-forward error back-propagation artificial neural network (ANN) simulation (Loukachine and Loeb 2003; Loukachine and Loeb 2004). This occurs when the total fraction of unknown cloud properties over the footprint, as defined by Eq. (2) of Loeb et al. (2003a)is greater than 0.35. The ANN has been trained using CERES Terra SSF data to provide a mapping between the CERES radiances and ADM-derived anisotropic factors over different surface types (ocean, land, desert, and snow). Validation tests show that the root-mean-square (rms) difference between instantaneous SW TOA fluxes from the ANN and original ADMs is approximately 9% for SW, 3.5% for LW daytime, and 3% for LW nighttime (WN rms differences are similar). Globally, approximately 5% of CERES TOA fluxes are inferred using the ANN scheme. The frequency of ANN use is significantly higher in mountainous regions, in coastal areas, and over snow/sea ice, where uncertainties in imager-derived cloud properties are larger. ANN is also frequently used at oblique CERES viewing zenith angles in the cross-track direction because MODIS is limited to cross-track viewing zenith angles that are smaller than 63°.

7. Summary

To determine the earth’s radiation budget from CERES, measured radiances at a given sun–Earth–satellite configuration must be converted to outgoing reflected solar and emitted thermal TOA radiative fluxes. CERES SW, LW, and WN ADMs are derived from 24 months of global CERES Terra radiances, imager-derived cloud parameters from MODIS, and meteorological information from the Global Modeling and Assimilation Office (GMAO)’s Goddard Earth Observing System Data Assimilation System (DAS) V4.0.3 product. The ADM scene types are defined as a function of scene parameters that have a strong influence on the anisotropy (or angular variation) of the earth’s radiation field at the TOA.

For clear scenes over the ocean, CERES Terra SW ADMs are defined as a function of wind speed and a theoretical correction is used to account for aerosol optical depth variation. Over land and desert, clear ADMs are defined for 1° latitude × 1° longitude equal area regions with a temporal resolution of 1 month. The ADMs are inferred from eight-parameter nonparametric fits to the bidirectional reflectance distribution function at these time and space scales. ADMs for clear scenes over snow/ice surfaces are stratified according to whether the surface is over permanent snow, fresh snow, or sea ice. Each of the three surface types are further stratified into “bright” and “dark” subclasses using predetermined monthly regional snow maps that classify all 1° × 1° regions with snow/sea ice as either bright or dark. Shortwave ADMs under cloudy conditions over the ocean are defined as continuous functions of a cloud parameter determined from imager-based cloud fraction and cloud optical depth. A sigmoidal fit is used to provide a continuous mapping between the cloud parameter and CERES radiances in each 2° angular bin interval in solar zenith angle, viewing zenith angle, and relative azimuth angle. Separate SW ADMs for liquid water, mixed phase, and ice clouds are derived from the sigmoidal fits. A similar approach is used to develop SW ADMs over land and desert, with additional approximations to account for the anisotropy of the underlying surface. ADMs for clouds over snow/ice are defined for discrete classes of cloud fraction and cloud optical thickness.

In the LW and WN regions, ADMs under cloud-free conditions are defined for one ocean class, five land categories corresponding to groupings of major IGBP surface types, and one snow class. The ocean and land clear-sky ADMs are further stratified into discrete intervals of precipitable water, vertical temperature change, and imager-based surface skin temperature. Over snow, clear-sky ADMs are stratified by surface skin temperature. When clouds are present over the ocean, land, or desert, the scene-type dependence of LW and WN radiances is represented by a parameterization that is a function of precipitable water, surface and cloud-top temperature, surface and cloud emissivity, and cloud fraction. Over snow, LW and WN ADMs are defined as a function of cloud fraction, surface skin temperature, and the temperature difference between the surface and cloud top.

In Part II of this paper, SW, LW, and WN TOA fluxes derived from the CERES Terra ADMs are assessed through extensive validation tests similar to those described in Loeb et al. (2003b). TOA fluxes from the new Terra ADMs will also be compared with TOA fluxes from the CERES TRMM ADMs and with fluxes based on algorithms developed during the Earth Radiation Budget Experiment (ERBE) (Smith et al. 1986; Suttles et al. 1992).

Acknowledgments

This research was funded by the Clouds and the Earth’s Radiant Energy System (CERES) project under NASA Grant NAG-1-2318.

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Footnotes

Corresponding author address: Dr. Norman G. Loeb, Mail Stop 420, NASA Langley Research Center, Hampton, VA 23681-2199. Email: n.g.loeb@larc.nasa.gov