## 1. Introduction

When integrated over the thickness of a cloud, microphysical properties such as particle phase, shape, and effective size determine the cloud optical depth and ice or liquid water path. These parameters govern the emittance and reflectance of the cloud, and, therefore, ultimately impact the radiation budget and influence climate. The need for understanding the global variability of cloud microphysical properties and their radiative effects has led to the development of several remote sensing techniques for deriving them from satellite-observed radiances. Interpretation of the satellite measurements relies on radiative transfer model calculations to relate a particular set of cloud microphysical characteristics to radiances entering and leaving the top of the atmosphere (TOA) from a cloud in a particular direction. The models provide the critical link between the radiance and a physical property.

A variety of spectra have been used to derive cloud microphyiscal properties. Inoue (1985) showed that the differences between the 11.0- and 12.0-*μ*m brightness temperatures observed by the Advanced Very High Resolution Radiometer (AVHRR) could be used to estimate the emittance of optically thin cirrus clouds. The errors in the emittance depended on the sizes of the cloud particles. Parol et al. (1991) found that the 11–12-*μ*m brightness temperature differences are related to the particle size for thin clouds. They also determined that particle shape affects cloud effective emittance. Arking and Childs (1985) developed a method that used the 0.67-, 3.7-, and 11-*μ*m AVHRR data with models based on Mie calculations of radiance to derive cloud fraction, optical depth, and cloud temperature and to assign a microphysical model. Coakley et al. (1987) and Radke et al. (1989) used the greater 3.7-*μ*m reflectivity of smaller droplets to detect the presence of ship tracks in boundary layer marine cloudiness. Stone et al. (1990) demonstrated a technique to derive optical depth using paired 3.73- and 10.8-*μ*m AVHRR radiances or matched 3.95- and 12.7-*μ*m Geostationary Operational Environmental Satellite (GOES) data. They inferred particle size by comparing the radiances to calculated brightness temperature differences between the two channels. Han et al. (1994) used near-nadir data from the 0.67-, 3.73-, and 10.8-*μ*m channels on the AVHRR to derive the effective droplet radii of warm clouds (temperature greater than 0°C) during daytime to provide the first estimates of cloud droplet sizes on a near-global scale.

Until recently, cirrus clouds were assumed to consist of ice spheres in radiative transfer modeling. Cirrus was generally characterized using Mie scattering phase functions, although infinitely long ice cylinders were occasionally invoked to describe cirrus scattering (e.g., Liou 1973). With the availability of scattering phase functions for particles shaped more like ice crystals (e.g., Takano and Liou 1989), it is possible to more accurately model radiances reflected from cirrus clouds. For example, Minnis et al. (1993a) showed that a radiative transfer model employing phase functions for randomly oriented hexagonal ice columns provided a more accurate depiction of the radiance field over cirrus clouds than Mie spheres. They also developed improved parameterizations of infrared cloud emittance by including scattering effects. Baum et al. (1994) demonstrated that radiative transfer calculations of 3.73-, 10.8-, and 12.0-*μ*m emittances using liquid-water spheres and randomly oriented hexagonal ice columns could explain the AVHRR-observed radiances from single-layer cirrus and stratus and from overlapping cirrus and stratus. Comparison of the calculations and the brightness temperature differences showed that the phase of some midlevel clouds could be determined by ensuring consistency between the data and models for all three channels. While these various efforts are beginning to show improvements in the retrievals of cloud microphysical properties, the advances come at the expense of more complexity and computation time that may limit the operational application of such techniques.

To avoid performing detailed radiative transfer calculations for each satellite scene, it is necessary to reduce the computation time using parameterizations while minimizing loss of accuracy. Rossow et al. (1988) employed a large set of lookup tables describing the angular distribution of reflected visible radiances for clouds having droplets with an effective radius of 10 *μ*m. The lookup tables were computed for clouds having a range of optical depths placed at several different levels in the atmosphere over surfaces with a wide range of albedos. Duplication of such tables for a variety of effective particle sizes greatly increases the demand for computer memory and access time. To minimize the computer demands and allow for variable particle radiative properties, Minnis et al. (1993b) developed a reflectance parameterization that interpolates between values in lookup tables of cloud visible reflectance. Their parameterization is applicable over any low-albedo surface at any altitude. They also developed a simple parameterization to describe the effective emittance as a function of cloud particle distribution and viewing zenith angle. Although their model incorporated infrared scattering by cloud particles, it was not capable of computing effective emittances exceeding unity. Thus, clouds with moderate optical depths or thin clouds at higher viewing angles could not be properly characterized. Ou et al. (1993) developed a relatively simple parameterization of 3.73-*μ*m emittance based on the 10.8-*μ*m emittance for a hexagonal ice-crystal size. However, the accuracy of that method is unknown.

A set of well-understood models that are spectrally consistent and complete is needed for simulating and interpreting cloud radiance fields. This paper addresses that need. Model calculations are performed to construct a set of lookup tables for visible (VIS) and solar-infrared (SI; 3.73 and 3.90 *μ*m) reflectance for a wide range of particle sizes. Various interpolation techniques are tested to determine the best method for estimating reflectance between node points. Effective emittance is also computed for the same particle distributions using a range of realistic cloud and clear-sky temperatures for several wavelengths commonly used by operational meteorological satellite imagers. These emittances are parameterized using a simple polynomial function that yields more realistic and accurate radiances than previous models. Errors are estimated relative to the detailed radiative transfer calculations for both the reflectance and emittance models. Examples are given to demonstrate the utility of these models in satellite analyses and simulations.

## 2. Radiative transfer model and data

The adding–doubling (AD) radiative transfer models described by Minnis et al. (1993a) were used to compute reflectance and effective emittance for a variety of clouds at selected wavelengths. The spectra correspond to channels on current operational meteorological satellites. Refractive indices, *m,* of ice and liquid water used in the calculations were estimated for the AVHRR channels 3 (3.75 *μ*m), 4 (10.7 *μ*m), and 5 (11.9 *μ*m) and for the *GOES-8* Imager channels 2 (3.90 *μ*m), 4 (10.8 *μ*m), and 5 (12.0 *μ*m) by convolving *m*_{λ} from Hale and Querry (1973) and Warren (1984) with the filter functions for each channel. Channel 4 on both instruments is referred to as infrared (IR), while channel 5 is denoted as split window (WS). Mean filter functions were computed for the AVHRR channels using the values for *NOAA-9, 10, 11,* and *12.* The GOES-Imager and AVHRR filter functions were obtained from NOAA (P. Menzel 1994, personal communication) and from Kidwell (1995), respectively. These filter functions are shown in Fig. 1a for the VIS and SI wavelengths and Fig. 1b for the IR and WS wavelengths. The index of refraction for wavelength *λ* = 0.65 *μ*m was used for the *GOES-7* visible and for the AVHRR and *GOES-8* channel 1 because the real part of *m* varies slowly and the imaginary part is essentially zero for the wavelengths within those channels. The SI radiation sensed by a satellite may arise entirely from emission by the scene or it may also contain a reflected solar radiation component. At the longer SI wavelengths, the solar constant decreases with increasing wavelength (Iqbal 1983) so that the spectral weighting of the filter function is influenced by the relative value of the solar constant. Thus, the weighting for the emitted component can differ from the solar component. Rather than using two different indices of refraction for each SI wavelength, a single value is computed by averaging *m* computed from the uniform- and solar-weighted filter-function convolutions. The resulting indices of refraction for each channel are listed in Table 1. Both the real and imaginary parts of *m* for ice at the WS wavelengths are greater than their IR counterparts suggesting less backscattered and more absorbed radiation for the IR channels. The difference between the WS and IR values of *m*_{i}(ice) is approximately double that for *m*_{i}(water). For the SI wavelengths, *m*_{i}(ice) is 100% greater than *m*_{i}(water).

### a. Cloud microphysics and optical properties

The AD computations were performed for both cirrus and water-droplet clouds. The latter are represented by Mie scattering optical properties computed using the program of Wiscombe (1980) for effective radii; *r*_{e} = 2, 4, 6, 8, 12, 16, and 32 *μ*m based on the modified gamma distribution (Hansen and Travis 1974) with an effective variance of 0.1.

Cirrus optical properties are based on the ray-tracing results of Takano and Liou (1989) for various combinations of randomly oriented hexagonal ice crystals having length-to-width ratios, *L*/*D,* of 20 *μ*m/20 *μ*m (C20), 50 *μ*m/40 *μ*m (C50), 120 *μ*m/60 *μ*m (C120), 300 *μ*m/100 *μ*m (C300), and 750 *μ*m/160 *μ*m (C750). In addition, new ray-tracing results for very small ice crystals, 5 *μ*m/5 *μ*m (C5), 10 *μ*m/5 *μ*m (C10.5), and 10 *μ*m/10 *μ*m (C10), are also considered. The spheroidal parameterizations of Takano et al. (1992) were used to estimate some or all of the optical properties for smaller ice crystals at infrared wavelengths because their size parameters are well below the geometric optics limit. The single-scattering albedos *ω̃*_{0}*Q* were computed with the Takano et al. (1992) parameterizations at 10.8 and 11.9 *μ*m for all of the ice crystals. Ray tracing was used for C300 and C750 to derive the phase functions and asymmetry parameters *g* at these wavelengths. The spheroidal parameterization was used to compute *g* for the other sizes. The phase functions were then computed using the Heyney–Greenstein (HG) formula (Heyney and Greenstein 1941). Because their aspect ratios are equal or close to unity, the average values for oblate and prolate spheroids having aspect ratios of *υ* = 0.5 and 2.0, respectively, were used for C5, C10, C20, and C50. Except for values of *Q,* the ray-tracing results were used at the SI wavelengths for crystals C20 and larger. Solar–infrared single-scattering albedos were computed with the spheroidal approximations for crystals smaller than C50. The phase functions and subsequent values of *g* for these same crystals were determined by averaging the ray-tracing and parameterization-based HG phase functions. The directly transmitted component *f*_{d} arising from the ray-tracing calculations (Takano and Liou 1989) was halved in these cases. For consistency, the Takano et al. (1992) spheroidal and large-crystal parameterizations were used to estimate *Q* for the visible channel. The other parameters for the VIS channel were taken from previous ray-tracing results at 0.55 *μ*m (Takano and Liou 1989). The 0.55-*μ*m phase function and single-scattering albedo can be substituted for the VIS because the real indices of refraction at both wavelengths differ by less than 0.2% and their imaginary indices are insignificant (Warren 1984).

The resulting single-crystal optical properties are given in Tables 2, 3, and 4 for the VIS, SI, and IR channels, respectively. At the VIS wavelengths, *Q* decreases with increasing *D*_{e}, while *g* generally increases with *D*_{e}. The exceptionally large value of *g* for C10.5 results from the large aspect ratio (*L*/*D* = 2), demonstrating the impact of shape on the scattering patterns. Its value also deviates from the trend in *g* seen for the SI data. The peak in *Q* for C10.5, however, is due more to its size parameter (*πD*_{e}/*λ*) than to its shape. For the IR and WS wavelengths, *g* varies monotonically. However, *Q*_{WS} peaks at C20 where the size parameter is ∼5, while *Q*_{IR} is more monotonic with a slight peak at C120 due to the transition from small to large crystal approximation. The single-scattering albedos, which are critical for particle size determination, are essentially monotonic with *D*_{e} for all of the considered spectra.

*N*is the normalized number of particles per unit volume in a given size bin and

*L*

_{1}and

*L*

_{2}are the respective lengths of the smallest and largest crystals in a size distribution. Several additional distributions were developed to extend the size range from contrail-sized to very large crystals and to provide a smoother change of cloud properties with effective particle size. The physical characteristics,

*D*

_{e}, mean volume

*V,*and cross-sectional area

*A*, of the resultant effective particles for the integrated size distributions are given in Table 5. The size distributions include those for cold cirrus (CC), cirrostratus (CS), and cirrus uncinus (CU) given by Takano and Liou (1989) and a warm cirrostratus (WCS) distribution that is an average of the warm cirrus and CS distributions from Takano and Liou (1989). The NOV and OCT distributions refer to data taken 1 November and 22 October 1986 during the First International Satellite Cloud Climatology Regional Experiment (FIRE) First Cirrus Intensive Field Observation (Cirrus IFO-I). Distributions derived from the results of Heymsfield and Platt (1984) for cirrus at temperatures

*T*= −40° and −60°C are designated T40 and T60, respectively. The T40, T60, NOV, and OCT distributions were discussed by Ou et al. (1993). A contrail cirrus (CON) distribution was developed from the data of Poellot and Henderson (1994) based on FIRE Cirrus IFO-II aircraft measurements taken in a contrail embedded within a cirrus cloud during a 22 November 1991 flight. Two other distributions were established to represent the extremes of the ice crystal sizes: a new contrail (NCON) comprising only particles smaller than C20 and a large-particle cirrus (LPC). These last three discrete distributions are given in Fig. 2. The C10.5 particles are included in the NCON distribution to make it more consistent with the other distributions that contain some crystals with significant aspect ratios.

*τ.*No subscript is used for the visible optical depth because it will serve as the reference optical depth for all of the calculations. In (2), the liquid water density

*δ*

_{liq}= 1.0 g cm

^{−3}. Similarly, using the discrete ice crystal distributions, the ice water path of the model cirrus clouds is

*N*

_{oi}is the normalized number of crystals per unit volume in given size bin

*i*represented by a particular hexagonal crystal of size

*L*/

*D.*The density of ice

*δ*

_{ice}= 0.9 g cm

^{−3}. A regression fit to the results of applying (3) to all of the distributions gives the following formula for ice water path in g m

^{−2}:

*D*

_{e}= 5–135

*μ*m. It should not be applied outside of this range.

Figures 3a and 3b show the phase functions for a range of ice cloud models for *λ* = 0.65 and 3.75 *μ*m, respectively. For both wavelengths, the largest particles have more forward and less side and backscattering than the smaller crystals. The scattering is nearly identical for scattering angles Θ > 30° for the smaller VIS models. The 22° and 46° halos seen in the VIS phase functions are replaced by a single halo at 30° for *λ* = 3.75 *μ*m that nearly disappears for the NCON distribution. The location and magnitude of the halos are complicated functions of crystal shape, orientation, and size parameter. Halos and other sharp features tend to diminish as the shape becomes more amorphous (e.g., Macke et al. 1996) and as the size parameter decreases. Thus, the SI ice crystal phase functions are generally much flatter than their VIS counterparts. The Mie scattering phase functions (Fig. 4) show a more complex variation with particle size for the two wavelengths. The visible Mie phase functions (Fig. 4a) are characterized by the extensive minimum at scattering angles, 80° < Θ < 130°, and the rainbow maximum near Θ = 138°. Both features are more pronounced for the larger droplet size. The Mie scattering phase functions for 3.75 *μ*m (Fig. 4b) generally have a less pronounced minimum in the sidescattering directions. The features tend to become smoothed as the size parameter decreases. Forward scattering increases with particle size for both wavelengths. Hansen and Travis (1974) provide a detailed discussion of Mie scattering.

The variations of the optical properties with effective particle size for all channels are listed in Tables 6, 7, and 8. The VIS values for *ω̃*_{0}*D*_{e} can be obtained by excluding CS and NOV from the parameterization. The single-scattering albedos for the IR and WS channels increase with increasing particle size while they decrease with increasing particle size for the VIS and SI wavelengths. At highly absorbing wavelengths, *ω̃*_{0}*ω̃*_{0}

Despite having larger values of *m*_{i}, the WS values of *ω̃*_{0}*ω̃*_{0}*μ*m water-droplet *m*_{r} with the 11.9-*μ*m value (1.258) of *m*_{r} doubles *ω̃*_{0}*m*_{r} (Hansen and Travis 1974), the opportunity for absorption decreases with increasing *m*_{r}. For water droplets, *m*_{r}(11.9 *μ*m) < *m*_{r}(10.8 *μ*m) yielding the expected low single-scattering albedos for 11.9 *μ*m. Thus, both parts of the index of refraction must be considered when using it to estimate the changes in single-scattering albedo. The retrieval of particle size is based on the variation of all three parameters, especially *ω̃*_{0}

### b. Reflectance and emittance computations

*ρ*at a particular set of solar zenith, viewing zenith, and relative azimuth angles,

*θ*

_{0},

*θ,*and

*ψ,*respectively, is one parameter of interest for satellite retrievals of cloud properties. Thus,

*ρ*was computed with the AD model for

*λ*= 0.65

*μ*m using 41 Gaussian quadrature points at

*ψ*= 0°, 5°, 15°, 30°, 45°, . . . , 150°, 165°, 175°, and 180° for the optical depths

*τ*= 0.25, 0.5, 1, 2, 3, 4, 8, 16, 32, 64, and 128. Linear interpolation was then used to construct a regular array of 11 zenith angles using

*μ*

_{0}= 1.0, 0.95, 0.85, . . . , 0.05 and

*μ*= 1.0, 0.9, 0.8, . . . , 0.1, where

*μ*

_{0}= cos

*θ*

_{0}and

*μ*= cos

*θ.*Reflectances were computed in a similar fashion at

*λ*= 3.75 and 3.90

*μ*m, but only for

*τ*⩽ 32, because

*ρ*is essentially constant at greater optical depths for most particle sizes. The interpolated, computed reflectances were compiled in lookup tables that require approximately 0.25 and 0.15 megabytes of computer storage per particle size for the VIS and SI channels, respectively. The directional albedo

*α*(

*μ*

_{0}) and spherical or diffuse albedo

*α*

_{d}were also calculated by integrating

*ρ*over the viewing angles and

*α*(

*μ*

_{0}) over the solar zenith angles, respectively. In this parameterization, the VIS optical depth serves as the reference optical thickness. The optical depth for a given wavelength is related to the visible optical depth by

*λ*) was determined by using the AD model to compute upwelling radiances,

*B*

_{λ}(

*T*) for 0° ⩽

*θ*< 72°, where

*B*is the Planck function. The computations were performed for surface temperatures

*T*

_{g}= 240, 260, 280, 300, 320 K; water-cloud temperatures

*T*

_{c}= 240, 255, 265, . . . , 295 K; and ice-cloud temperatures

*T*

_{c}= 195, 210, . . . , 270 K. The radiances were computed for spectral optical depths corresponding to the VIS optical depths

*τ*= 0.25, 0.5, 1, 2, 3, 4, 8, 16, and 32 using (5) to convert from

*τ*to

*τ*

_{λ}. In the calculations, it was assumed that the surface is black and all atmospheric moisture is confined to the subcloud layer. This assumption produces a negligible error in application because surface albedos are small at these wavelengths. Thus, any multiple scattering between the cloud and surface will yield upwelling radiances that are much smaller than the surface-emitted radiance. The model is formulated in terms of the upwelling radiation at cloud base whether it emanates from the atmosphere, the surface, or a combination of the two. A nominal atmospheric optical depth was determined based on the layer temperature and the wavelength using a relatively moist atmosphere. Several cases were also evaluated using totally dry atmospheric layers. Clear-sky equivalent blackbody temperatures

*T*

_{s}were computed for each channel using the specified atmospheric optical depths and effective atmospheric temperatures. The air temperature closest to the average of

*T*

_{c}and

*T*

_{g}and the corresponding optical depth were used to represent the subcloud layer for a given case. The effective emittance for channel

*i*is

_{i}

*B*

_{i}

*T*

*B*

_{i}

*T*

_{s}

*B*

_{i}

*T*

_{c}

*B*

_{i}

*T*

_{s}

_{ai}

*ω̃*

_{oi})

*τ*

_{i}/

*μ*]

_{a}depend mostly on

*ω̃*

_{0}

## 3. Results and analysis

### a. Reflectance

*μ*

_{0}= 0.65 and

*τ*= 1 for

*r*

_{e}= 6 and 16

*μ*m, and for T60 and OCT for

*λ*= 0.65 and 3.75

*μ*m, respectively. Similarities in the reflectance patterns of different ice particle sizes at a given wavelength follow from the commonalities in the phase functions (Fig. 3). Variations in the 0.65-

*μ*m reflectance patterns for the water droplets (Figs. 5c,d) arise from phase function differences. For example, the rainbow peak at Θ = 138° is more pronounced for larger droplets while the backscatter maximum is diminished (Fig. 4a). At

*r*

_{e}= 16

*μ*m, a relative reflectance maximum appears at

*θ*= 65°,

*ψ*= 135° in Fig. 5d, but not at

*θ*= 53°,

*ψ*= 180°, where a distinctive maximum occurs for

*r*

_{e}= 6

*μ*m (Fig. 5c). Even more striking reflectance differences between the two sizes are conspicuous at 3.75

*μ*m. The differences between the droplet and ice crystal VIS reflectance patterns are also significant and have been noted in previous studies (e.g., Minnis et al. 1993a). Discrepancies between the ice and water reflectances are also quite noticeable in the SI reflectance patterns (Fig. 6) although there are substantial differences between SI and VIS reflectance patterns for the same effective particle size.

Integration of the bidirectional reflectances yields the albedos. Figures 7 and 8 show examples of albedo as a function of *μ*_{0} for the models in Figs. 5 and 6, respectively, at various optical depths. The small-particle albedos are typically greater than those for the larger particles of both phases. Water droplets reflect more than ice crystals at 3.75 *μ*m because of the larger values of *ω̃*_{0}*μ*m because *g* is generally smaller for the crystalline shape than for the spheres. Albedos at 0.65 *μ*m exceed those at 3.75 *μ*m because of the latter’s smaller values of *ω̃*_{0}*μ*_{0} is less than the corresponding increase in the VIS albedo.

The phase and particle size dependencies are more evident in the variations of diffuse albedo with optical depth shown in Figs. 9 and 10 for 0.65 and 3.75 *μ*m, respectively. Except for *r*_{e} = 2 *μ*m, the diffuse VIS albedo (Fig. 9) for a water droplet cloud at a given optical depth is less than that for most ice clouds. At *r*_{e} = 2 *μ*m, *α*_{d} is similar to that of a cloud composed of medium-sized ice crystals (T40). Conversely, most of the ice-cloud SI albedos (Fig. 10b) are less than those for water-droplet clouds (Fig. 10a) at a given SI optical depth. Exceptions are the contrail distributions that produce albedos comparable to the 6- and 12-*μ*m water droplets. More typical cirrus clouds have SI albedos that are less than those for *r*_{e} = 16 *μ*m. There is overlap between the diffuse albedos for droplets smaller than 8 *μ*m for *τ*_{3.75} < 3 (*τ* < 4). This overlap results in the well-known nonmonotonic variation of reflectance with droplet size that makes the retrieval of *r*_{e} ambiguous for methods using the SI data (e.g., Han et al. 1994). It results from the location of the extinction efficiency peak near *r*_{e} = 4 *μ*m (Table 8). At larger optical depths, the albedo overlap disappears because increasing absorption due to the monotonically decreasing values of *ω̃*_{0}

### b. Emittance

*r*

_{e}= 6

*μ*m are shown in Figs. 11a and 11b, respectively, as functions of log(

*τ*/

*μ*). There is considerable variability in these results, especially for channel 3 as a result of increased scattering. There are many points having ε > 1. This effect, in most cases, is confined to large optical depths, large viewing zenith angles, and low contrast between the cloud and the surface. Fits to the model given by Minnis et al. (1993a),

*a*

*τ*

*μ*

^{b}

*a*and

*b*are regression coefficients, using only data with ε < 1, are also shown for comparison with the data. Although many of the data points fall on the curve, there are a substantial number that do not follow this simple relationship. It is obvious that the simple functions traditionally used to fit the data (e.g, Minnis et al. 1993b) cannot include all of the points, in particular those having ε > 1.

To examine the variability not explained by *τ*/*μ,* the emittances are replotted in Fig. 12 as functions of the temperature difference, Δ*T*_{sc} = *T*_{s} − *T*_{c}. The channel-3 effective emittances (Fig. 12a) have maximum values of ∼2.9, while the IR emittances (Fig. 12b) peak at ∼1.7. Greater scattering at 3.75 *μ*m gives rise to the larger effective emittances. Effective emittance increases with decreasing Δ*T*_{sc} in an almost logarithmic fashion, especially for channel 3. The horizontal spread in a given string of points is due to the change in *μ.* For the dry atmospheres, there is no horizontal spread, so the strings form vertical lines at particular differences (e.g., Δ*T*_{sc} = 30 K in Fig. 12a). For the largest values of Δ*T*_{sc}, ε < 1 for optical depths as large as 16 or 8 at 3.75 (Fig. 12a) and 10.8 *μ*m (Fig. 12b), respectively. Because of this apparent logarithmic dependence, values of Δ*T*_{sc} < 4 K were not used in the parameterization. As shown later, large emittance errors for low thermal contrast have minimal impact on the derived temperature.

*T*

_{sc},

*T*

_{s},

*T*

_{c},

*τ,*and

*μ,*revealed that functions of

*T*

_{s}, Δ

*T*

_{sc},

*τ,*and

*μ*were most highly correlated with ε for a given particle size. For example, for

*r*

_{e}= 6

*μ*m at

*λ*= 3.75

*μ*m, the linear correlation coeffcients for ε with

*T*

_{c}, 1/ln(

*T*

_{s}), 1/ln(Δ

*T*

_{sc}),

*τ,*and

*μ*are 0.21, −0.14, 0.49, 0.60, and −0.21, respectively. When

*T*

_{s}and Δ

*T*

_{sc}are included together in a multiple regression,

*T*

_{c}becomes an insignificant variable in the regression. Except for

*τ,*the correlations decrease with increasing

*r*

_{e}. Han (1992) also found significant dependence of the 3.75-

*μ*m emittance on surface and cloud temperatures. At 10.8

*μ*m, the respective coefficients are 0.06, −0.05, 0.13, 0.66, and −0.26. To account for these dependencies and minimize the errors, multiple regression was performed for each of the nine discrete values of

*τ*using the formula

*ζ*= 1/ln(Δ

*T*

_{sc}) and

*ξ*= 1/ln(

*T*

_{s}). The coefficients

*d*

_{ijk}were determined by minimizing the squared error. These regression analyses generated nine sets of 30 coefficients for each microphysical model and spectral band. Emittances for optical depths off the nodes are determined by four-point Lagrangian interpolation in ln

*τ*for all interior values and linear interpolation for

*τ*< 0.25 and

*τ*> 16. If

*τ*> 32, ε = ε(

*τ*= 32). To further reduce errors, Lagrangian interpolation is used for 16 <

*τ*< 32 if the difference in ε between optical depths of 16 and 32 is greater than 1%. For a given set of conditions, ε is found with the parameterization that consists of (9) and the interpolation system. A four-parameter regression fit using an additional polynomial in ln

*τ*was also examined, but it did not produce a sufficiently accurate fit for the SI wavelengths.

The rms or standard errors of the estimates (SEE) for the regression fit to (8) shown in Fig. 11a for the 6-*μ*m water droplets at 3.75 *μ*m are 0.363 and 0.140 for all of the data and for ε < 1, respectively. The corresponding errors are reduced to 0.014 and 0.012 with (9). For the 10.8-*μ*m curve in Fig. 11b, the respective SEEs are 0.084 and 0.049 compared to 0.002 and 0.002 for the fit to (9). Values for SEE in absolute emittance are plotted for all of the liquid and ice particle models in Fig. 13 for channels 3, 4, and 5. The results for the corresponding *GOES-8* spectra are not shown because they are nearly identical to those for the AVHRR channels. The greatest SEEs occur for the small particles at SI wavelengths where *ω̃*_{0}

An effective technique for demonstrating the variability of emittances is to plot the brightness temperature difference, BTD_{λ} = *T*_{λ} − *T*_{IR} or *T*_{IR} − *T*_{λ}, as a function of the IR temperature *T*_{IR}. Figure 14 shows the BTDs computed using the 3.75- and 11.9-*μ*m parameterizations for a range of *r*_{e} and *D*_{e} at *θ* = 30°, *T*_{g} = 295 K, and *T*_{c} = 260 K for *τ* ⩽ 16. The water-droplet BTDs cover a greater range at a given value of *T*_{IR} than the corresponding ice-crystal values for both wavelengths. Thus, there is more sensitivity in the BTDs to changes in droplet size than to variations in ice-crystal size. At 11.9 *μ*m (Figs. 14c,d), the BTD increases with decreasing particle size for both phases. Similar behavior is evident for most of the SI ice-crystal distributions (Fig. 14b). Conversely, most of the SI water-droplet BTDs (Fig. 14a) increase with increasing effective radius. Both phases have a nonmonotonic variation in BTD at 3.75 *μ*m with the maximum BTDs occurring at *r*_{e} ≈ 16 *μ*m and *D*_{e} ≈ 24 *μ*m. At greater clear-cloud temperature contrasts, the ice-crystal BTD maximum shifts to smaller sizes. Thus, at colder cloud temperatures, it is possible to obtain a monotonic range in *D*_{e} from 18 to 135 *μ*m. The BTDs are less than or equal to zero for *τ* = 16 for all of the models except for *r*_{e} = 2 and 4 *μ*m at the IR and WS wavelengths. Because of the extremely small extinction efficiencies at these wavelengths, the emittances are less than unity at *τ* = 16. Thus, the 10.8- and 11.9-*μ*m temperatures do not reach *T*_{c} for *r*_{e} = 2 *μ*m.

Given only SI and IR data, it would be difficult to unambiguously determine particle size when *r*_{e} exceeds ∼10 *μ*m or when *D*_{e} < 50 *μ*m in the case of Fig. 14. Because the particle sizes corresponding to the maximum BTDs vary somewhat with Δ*T*_{sc}, the range of potentially ambiguous particle sizes will also vary with Δ*T*_{sc}. The monotonic changes in Figs. 14c and 14d may be used together with the SI BTDs, however, to resolve both phase and particle size in this particular case. This nonmonotonic behavior may explain why contrails are only occasionally detectable with the SI data, but are often seen in images of BTDs from 10.8- and 11.9-*μ*m data (e.g., Lee 1989). The overlap of the contrail-sized-particle BTDs with those for larger ice crystals should often make contrails indistinct from other cirrus clouds at the SI wavelengths at night. The addition of the significant solar reflection from small-particle contrails (e.g., Fig. 10) would render the contrails more distinguishable in certain conditions during the daytime. At 11.9 *μ*m, small-particle contrails should be more easily identified because of the monotonic variation of BTD with *D*_{e}.

### c. Daytime solar infrared

*B*

_{λ}

*T*

_{λ}

_{λ}

*B*

*T*

_{c}

_{λ}

*B*

*T*

_{b}

*ρ*

_{λ}

*μ*

_{0}

*E*

_{λ}

*δ*

*d*

*T*

_{b}is the equivalent blackbody temperature of the total upwelling radiance below the cloud,

*E*

_{λ}is the spectral solar constant,

*δ*is the normalized Earth–Sun distance, and

*d*is the day of the year. In the absence of atmospheric absorption, the upwelling radiance below the cloud is approximated as

*B*

*T*

_{b}

_{gλ}

*B*

*T*

_{g}

*α*

_{gλ}

*E*

_{λ}

*μ*

_{0}

*α*

_{λ}

*μ*

_{0}

_{aλ}

*μ*

_{0}

_{g}is the surface emittance,

*α*

_{g}is the surface albedo,

*α*

_{λ}is the cloud albedo, ε

_{aλ}is the spectral absorption emittance, and

*T*

_{g}is the skin temperature of the surface. At night or when

*α*

_{gλ}= 0, the first term in (11) is equivalent to

*B*(

*T*

_{s}) and

*B*(

*T*

_{b}) =

*B*(

*T*

_{s}). At night, therefore, (10) reduces to (6). In (11), the term in the brackets is the solar radiation reaching the surface; it is assumed to be diffuse. Because the reflected component is removed explicitly with the cloud albedo, the absorption emittance is used in this term. Secondary- and higher-order reflections between the cloud and the surface are neglected in this approximation.

## 4. Discussion and applications

### a. VIS reflectance interpolations

Application of the reflectance models to specific angles and optical depths requires interpolation between the lookup table values. Various interpolation methods were evaluated using the VIS reflectances calculated for a variety of angles and optical depths different from those used to create the lookup tables for the model. The AD calculations were performed for the CS and 8-*μ*m water-droplet models using random sets of viewing and illumination angles selected for *μ*_{0} > 0.15, *μ* > 0.34, and 0 ⩽ *ψ* ⩽ 180°. Ten randomly selected values of log*τ,* where log*τ* = −1, 2, were used to complete the calculations resulting in a total of 12 250 samples for each model. The interpolations for the CS and 8-*μ*m droplets should be representative for the ice and water-droplet models.

Reflectances were estimated from the lookup tables using nearest-node values and interpolations with various combinations of linear and Lagrangian methods. The results were compared to the AD values to determine the optimal technique. Table 9 summarizes the errors in estimated reflectance for a few of the techniques. The hybrid interpolation combines linear interpolation in *ψ* and ln*τ* with four-point Lagrangian interpolation in *μ*_{0} and *μ.* The hybrid method is comparable to the four-dimensional, four-point Lagrangian technique and is more accurate than linear interpolation that is actually log-linear in *τ* for *τ* > 0.25. Overall, the bias errors are −0.5% and 0.1% for the 8-*μ*m and CS models, respectively. The corresponding rms errors are 3.0% and 9.9% compared to 21.8% and 19.0% for the nearest node approach. Although the hybrid method is more accurate than the linear technique, it is computationally slower by a factor of 2. Thus, the accuracy of the linear method may be sufficient for a particular application. There appears to be no reason for using more computationally expensive four-point Lagrangian method instead of the hybrid technique.

To better understand the estimated reflectance errors, the means and extremes from the hybrid interpolation are plotted as a function of scattering angle in Fig. 15. For the 8-*μ*m model (Fig. 15a), the mean errors are greatest near 138° and 179° and minimal elsewhere. These error variations depend on the features of the phase function. The change in the phase function with Θ between 120° and 140° is greater than an order of magnitude for *r*_{e} = 8 *μ*m (e.g., Fig. 4a) and is highly nonlinear. Similarly, the phase function increases by a factor of 8 between 174° and 180°. Thus, interpolation between the relatively low resolution of the model nodes is insufficient to capture the shape of the curve at these important angles. The dependence of error on the phase functions is seen more clearly in the CS results (Fig. 15b), where the features of the CS phase function (similar to WCS in Fig. 3a) are mimicked in the mean reflectance errors. Here the greatest errors occur in the forward (Θ < 50°) and backscattering (Θ > 174°) directions. Except for the cross-scattering directions (60°–105°) in the CS model, the error extrema in Fig. 15 differ substantially from the means. Most of the errors are considerably smaller than the extrema. The largest relative differences primarily occur for very small optical depths where single scattering dominates.

These errors may be reduced by including more nodal points or by saving the Fourier coefficients from the azimuthal expansions used in the AD calculations instead of retaining single values at a given *ψ.* However, the zenith variations affecting the scattering angle could still cause large uncertainties if the number of nodes remained small. The size and complexity of the lookup tables must be weighed against memory and processing time constraints. Capturing all of the phase function features accurately in a lookup table will still be elusive, however, because of the various combinations of *μ*_{0}, *μ,* and *ψ* that can determine a given value of Θ. Due to generally smoother phase functions at 3.75 *μ*m (Figs. 3b and 4b), the reflectance interpolation errors will be smaller except when Θ > 174° for the ice models and when Θ > 135° for the 32-*μ*m water-droplet model.

Although these models may be used for a variety of applications, they will most often be applied to satellite observations. In practice, the overall interpolation errors may be better or worse depending on the particular viewing geometry of a given satellite. As shown in the appendix, bias and rms errors due to interpolation are smaller than the values in Table 9 for more than half and 75%, respectively, of the considered realistic satellite sampling scenarios. The average ice-crystal rms error is less than 3% for all of the satellite cases.

### b. Cloud water path and albedo

The VIS albedo varies by a maximum of ∼0.1 for a particular optical depth (Fig. 9) over the range of particle sizes considered here. The variation of *α*_{d} with *r*_{e} for a fixed value of *τ* is much less than that for a fixed value of cloud water path. Ackerman and Stephens (1987), Stephens and Greenwald (1991), and others have shown both theoretically and empirically that the broadband solar albedo can change dramatically with *r*_{e} for a fixed value of LWP. It is instructive, however, to understand the importance of particle size when relating the hydrological cycle to the VIS albedo or, consequently, the VIS radiance field. This relationship depends on how the albedo changes when a particular amount of water vapor is condensed or frozen in the atmosphere giving rise to a fixed liquid or ice water path.

Figure 16 demonstrates the sensitivity of the VIS albedo to particle size in terms of water path. The albedo can change by more than 0.70 for a given amount of liquid or ice water depending on the particle size distribution assumed in the calculation. For liquid water clouds (Fig. 16a), the albedo range for constant LWP exceeds 0.2 for LWP between 2 and 600 g m^{−2}. The same range in albedo occurs for IWP between 0.8 and 150. These wide ranges in albedo for a given cloud water path have serious implications for climate model calculations and for satellite-based retrievals of optical depth. For example, if a model calculation specifies LWP = 100 g m^{−2}, the VIS albedo can vary by 0.33 to 0.92 for effective droplet sizes between 32 and 2 *μ*m, respectively. An assumption of *r*_{e} = 10 *μ*m, for instance, converts the LWP to *a*_{d} ≈ 0.64. For the frequently observed effective droplet sizes between 4 and 16 *μ*m, this assumption could lead to albedo errors as great as 0.16. Conversely, the optical depth retrieved using the 10-*μ*m water-droplet assumption can often lead to errors in *τ* or LWP of more than 100% for commonly observed droplet sizes. Similar errors can also occur for cirrus clouds.

### c. Emittance and temperature errors

Minnis et al. (1993b) showed that (8) provides a more accurate representation of ε than (7). As demonstrated earlier, (9) decreases the uncertainties in ε estimated with (8) by a additional 100% or more. Thus, the errors from the earlier models are considerably greater than those for the present parameterization. Although computationally more intensive, the current technique provides more accuracy and operates over a greater range of optical depths than the earlier approaches to effective emittance. The simpler methods represented by (7) and (8) may be sufficiently accurate for the IR and WS bands in some applications. As the precision of remote sensing measurements increases, however, the higher accuracy of (9) will be required even for the less sensitive infrared window bands.

*R*=

*B*

_{λ}(

*T*) or equivalent blackbody temperature

*T*

_{λ}at the top of the atmosphere is ultimately the quantity of interest. Thus, it is necessary to consider the sensitivity of

*T*to errors in ε. Given the definition of

*T*in (6) and the Planck function, it can be shown that errors in

*T*due to the emittance errors can be estimated by

*R*=

*B*

_{λ}(

*T*

_{c}) −

*B*

_{l}(

*T*

_{s}). From (13), it is expected that if the emittance is small or

*T*

_{c}is close to

*T*

_{s}, then

*S*

_{Tλ|ε}

*T*

_{sc}(i.e., Δ

*R*) are large, then a minimal emittance error can cause a large temperature error.

Values of Δ*T* were computed as the differences between the original AD and corresponding parameterized temperatures for *r*_{e} = 8 *μ*m and T40. More than 88% and 95% of the differences are less than ±0.5 K for water droplets and ice crystals, respectively. Although they may be as large as −10 K occur in a few instances, errors less than ±1 K occur in 95% of the water-droplet cases. The rms errors for all of the models are summarized in Table 10 for the two SI wavelengths. When ε < 1, the errors are considerably smaller than for all cases indicating that the characterization of effective emittances greater than unity is difficult, even with the current approach.

The temperature errors are generally greatest for smaller particle sizes (Table 10). The errors are largest for the NCON distribution and decrease by a factor of 2 as particle size increases to the CC distribution. Except for contrail distributions, the water-droplet Δ*T*s are similar to the ice-crystal temperature errors. Because of the greater possible range in Δ*T*_{sc}, the sensitivity factors can be much greater for the ice-crystal than for the water-droplet models. Thus, Δ*T* for the ice models can be larger than that for the liquid models despite the greater emittance errors for the water droplets. In all cases, Δ*T* increases with *τ* (or ε) and Δ*T*_{sc}, as expected from (12) and (13). For channels 4 and 5, the temperature errors have negligible size dependence and no significant differences between all ε and ε < 1. At both IR and WS wavelengths, the rms temperature errors for water droplets and ice are ∼0.06 and 0.04 K, respectively.

The parameterizations given by (9), (11), and the interpolation technique were tested by comparing AD calculations that explicitly included both the solar and emitted SI components for an independent dataset (see the appendix). For nocturnal cases, the errors in simulated values of *T*_{SI} that would be observed by a satellite are approximately ±0.5 and 1.0 K for water-droplet and ice-crystal clouds, respectively. The latter error reduces to 0.6 K if temperature and emittance interpolations in optical depth are used for *τ* > 8 and *τ* < 8, respectively. During the daytime, the rms errors are approximately ±0.6 and 1.0 K, respectively, for water and ice if the optimal interpolations are used. The temperature errors can be as great as 10 K for some scattering angles and large cloud-surface temperature contrasts. The use of (11) in the parameterization of combined cloud and surface radiation introduces no significant errors in the simulated temperatures.

### d. Applications

The models and parameterizations developed here can be used in either simulations of radiance fields for specified cloud conditions or in the retrieval of cloud properties from multispectral radiances. Examples of the simulations were given in Fig. 14. Figures 17–20 show the AVHRR imagery and the results of the parameterization calculations plotted over two-dimensional histograms of the AVHRR BTDs. Data from the *NOAA-11* AVHRR are given in Fig. 17 for cirrus clouds over the Coral Sea east of Australia observed 17 January 1993. The VIS–IR histogram (Fig. 17b) is analyzed with the layer bispectral threshold method (LBTM; see Minnis et al. 1995), which divides the histogram into clear (box at dark, warm corner) and cloudy regions (remaining area). The three curves define the relationships between VIS counts, which are directly proportional to reflectance, and *T*_{IR} for clouds at the tropopause (gray, steepest line), 6-km (middle line), and 2-km (nearly horizontal line extending rightward from VIS count of 14). The 2-km line is defined using *r*_{e} = 10 *μ*m, while the other two are computed using the CS model. These lines would continue off the graph gradually becoming horizontal at the temperature for their particular altitudes (see Minnis et al. 1993b). The numbers on the plot indicate the frequencies of each pair of VIS count and *T*_{IR}. The LBTM analysis yields a high cloud with a center near 12 km with *T*_{c} = 237 K. Using the LBTM result, the modeled SI BTDs for ice particle sizes between 24 and 45 *μ*m plotted in Fig. 17c encompass most of the data suggesting an average value of *D*_{e} = 30 *μ*m. The WS BTDs for the same particle sizes also account for most of the data in Fig. 17d indicating that the parameterizations provide a consistent spectral characterization of the observations.

Figure 18 shows the results for a lower-level cloud observed in Fig. 17a from the *NOAA-11* AVHRR. The structure of the VIS–IR histogram (Fig. 17a) and the LBTM analysis yield a midlevel cloud deck at 3 km where *T*_{c} = 283.7 K. Using this result with a correction for water vapor absorption based on the correlated k-distribution technique (Kratz 1995) to compute BTDs yields the curves for *r*_{e} = 8–16 *μ*m that envelope most of the SI BTD data in Fig. 18b and some of the WS BTD values in Fig. 18c. The WS models do not account for all of the differences in optically thick cloud cases. The variation of *T*_{IR} for *BTD* ⩽ 0 K in Fig. 18c suggests that the cloud-top temperature actually varies from 282.5 to 285 K so that no single curve can account for all of the data.

*NOAA-14* IR data are shown in Fig. 19a for a multilevel case seen over the Gulf Stream east of Cape Hatteras, North Carolina, during 16 January 1997. A low-cloud layer is evident near 281 K (1.1 km) in the corresponding VIS-IR histogram in Fig. 19b. Higher clouds do not appear to form a distinct layer in the VIS-IR data. However, the WS BTD data in Fig. 19d suggest an upper-level cloud between 252 and 262 K. The closest sounding, taken from Wallops Island, Virginia, indicates a very moist layer under an inversion at 256 K. Curves outlining much of the data can be constructed if it is assumed that there are two layers with *T*_{c1} = 281 K and *T*_{c2} = 256 K over a clear-sky temperature of *T*_{s} = 292 K. Unlike a simple theoretical system, real multilayered clouds can present a very complex radiance picture. The curves overlaying the data in Figs. 19c and 19d represent only one scenario for this case. Curves A and B represent the BTDs for *r*_{e} = 8 and 12 *μ*m, respectively, for the low cloud. These models, which account for many of the warmer data points, result in a spread of BTD(SI−IR) ∼ 10 K for the lower cloud at 281 K. Although the corresponding curves in Fig. 19d do not appear to envelope many of the warmer data, the actual number of points within the curves is far greater than indicated by the discrete representation of the data. Furthermore, inclusion of the ∼1.5 K spread in the clear-sky BTD(IR−WS) in these curves would encompass many more of the warmer data.

The upper-level cloud in Figs. 19c and 19d is represented with a single particle size, T60 or *D*_{e} = 30 *μ*m, to illustrate the effects of a varying background. The value of T60 was selected because its curve (not shown), determined using *T*_{c2} over the clear background, bisected much of the colder data. Curves C and D in Fig. 19c were computed using the background radiances derived for an optically thin, low-cloud yielding *T*_{s} = 285 K and for the optically thick cloud at *T*_{s} = 281 K, respectively. The maximum BTD(SI−IR) at 285 K is ∼18 K and the minimum at 281 K is ∼10 K. These differences were used to estimate the clear-sky reflectance for use in (11). The results apparently account for much of the colder data. Using these two background temperatures in the calculations for BTD(IR−WS) in Fig. 19d provides a consistent scenario; a large portion of the colder data points are encompassed by C and D. The T60 curve for a clear background (dotted line in Fig. 19d) accounts for some of the larger BTDs. The outlying colder points with smaller BTD(IR−WS) and larger BTD(SI−IR) probably correspond to optically thicker clouds at temperatures different from 256 K. The remaining unexplained points are warmer than 275 K with BTDs greater than explained by the water-droplet models. These points may correspond to an optically thin high cloud near 220 K. The cloud mass to the east in Fig. 19a contains many pixels with *T* < 225 K. The Wallops sounding also indicates a very moist layer between 205 K (11.5 km) and 226 K (8 km). Except for the few outlying points, the parameterization results used here yield a consistent depiction of the observed data.

Farther north during 26 September 1996, contrails were prevalent over Virginia, the Chesapeake Bay, and the adjacent waters of the Atlantic. Figure 20a shows some of these contrails in a *NOAA-12* AVHRR IR image. The VIS-IR LBTM analysis (Fig. 20b) places these contrail clouds just below the tropopause at 11.4 km (*T*_{c} = 218 K). For this cloud temperature, both the SI (Fig. 20c) and WS (Fig. 20d) BTDs indicate that the mean particle size is ∼21 *μ*m. This particle size is only slightly greater than the CON model developed to represent an old contrail confirming, at least in this case, the smaller size of ice crystals in relatively new contrails.

The final example seen in Fig. 21a is for a cirrus cloud observed in 4-km *GOES-8* data taken at night over the north central Oklahoma Atmospheric Radiation Measurement (ARM) Program Central Facility (Stokes and Schwartz 1994) during 15 April 1995. The ARM micropulse lidar returns (J. D. Spinhirne 1995, personal communication, available online at http://virl.gsfc.nasa.gov/mpl.html) indicate that the cloud center was located at 10.5 km or 230 K. Model calculations were performed using *T*_{c} = 230 K and the observed clear-sky temperature. Figure 21b shows the best-fitting model results plotted over the SI BTDs for data over the lidar site. Both the ice-crystal models, CC and WCS, and the water-droplet models, 2 and 4 *μ*m, encompass most of the data. The water-droplet models, however, cannot explain the WS BTDs (Fig. 21c) that are also enveloped by the CC and WCS models. This consistency demonstrates the potential utility for using the WS data for phase selection.

These examples cannot fully represent the myriad of cloud conditions found in nature. Nor should it be concluded that the models can be used to accurately portray the radiances for any and all cloud systems. However, they demonstrate that the parameterizations can provide a realistic and spectrally consistent characterization of a variety of clouds in different viewing and illumination situations. Figures 17–21 show the models plotted over the data, but do not provide an objective retrieval of the cloud properties. The development of a methodology to retrieve phase, particle size, and optical depth is a complex process and beyond the scope of this study. The excellent agreement between the models and the data in these examples, though, show that the parameterizations derived here can be used in any retrieval method utilizing the AVHRR and GOES-Imager data.

### e. Other considerations

The models developed here are based on a highly idealized set of cloud conditions and cannot be assumed to represent all clouds. Because infinite, plane-parallel clouds are used in the model calculations, it is likely that there are many situations, such as those involving broken, scattered, or horizontally inhomogeneous clouds, in which these model results will not be applicable. Vertical cloud inhomogeneities in cloud particle size may also yield different radiances than those predicted from the parameterizations. Cloud ice particles can take a variety of shapes or orientations that may differ from the randomly oriented, hexagonal ice columns used here. The reflected radiance fields vary with changes in the ice particle orientation or shape changes (e.g., Takano and Liou 1989; Macke et al. 1996). The dramatic effect of particle shape on *g* was demonstrated earlier in the discussion about the C10.5 crystal. Given the phase function and other optical properties, the calculations performed here can be applied to any particle shape, size, or orientation. Hexagonal columns were used here because they are relatively common in cirrus clouds (e.g., Heymsfield and Platt 1984) and can occur as components in more complex ice crystals. Whether they are the best shape to characterize cirrus in general is an open question. Much more data analysis is required before it can be determined how accurately the present models represent actual cirrus. Future remote sensing techniques that determine both particle shape and size, however, would obviate the need for using a single shape in the cloud retrieval process. Conversely, additional knowledge of the distribution of ice particle shapes and sizes would permit more accurate simulations of the cirrus cloud radiance fields.

## 5. Concluding remarks

Global retrievals of cloud properties require an accurate, spectrally consistent, and computationally efficient representation of cloud radiances for all relevant viewing and illumination conditions observed from satellites. To address this need, parameterizations of reflectance and emittance for wavelengths commonly used for cloud property retrievals were developed and comprehensively evaluated in this paper.

The reflectance parameterization consists of a set of lookup tables covering a broad range of water-droplet effective radii and hexagonal ice crystal distributions for the visible and solar infrared channels on AVHRR and GOES. Values between the angle and optical depth nodes for a given particle size and optical depth can be most accurately and efficiently determined through a four-point Lagrangian interpolation in the cosines of solar and viewing zenith angles and linear in relative azimuth angle and the natural logarithm of optical depth. Heretofore, the errors in reflectance due to interpolation have not been quantified. For the best interpolation method used here, the interpolation errors for the VIS models are generally less than or equal to 3% and 5% for water droplets and ice crystals, respectively, for useful satellite viewing conditions. For solar–infrared reflectance, the interpolation errors are less than 3% and 2% for droplets and ice crystals, respectively. The errors are greatest near the backscattering direction. In most conditions, however, the reflectance errors are minimal for both the visible and solar–infrared wavelengths. Techniques for improving the interpolations were discussed.

The effective emittances for the solar–infrared and infrared–window wavelengths were parameterized with a multivariate polynomial in the cosine of viewing zenith angle, the clear-cloud temperature difference, and the surface temperature. The optical-depth dependence of effective emittance, which includes both the scattering and emission effects for a particular cloud particle size, is most accurately determined through four-point Lagrangian interpolation between optical depth nodes. Overall, the effective emittances for the infrared window and split-window computed with the more exact adding–doubling radiative transfer model can be reproduced to within ±0.4% using the current parameterization. The corresponding effective emittances for the solar infrared channels are generally within ±2% and ±1% of the adding doubling results. These new parameterizations yield an order of magnitude improvement over previous models. In addition, they account, for the first time, for effective emittances that exceed unity. The resulting rms temperature errors due to emittance uncertainties are 0.06, 0.04, and 0.85 K for the infrared, split-window, and solar–infrared wavelengths, respectively. During the day, reflection of the incoming solar radition at the solar–infrared wavelengths must also be taken into account when computing the effective blackbody temperature of a scene. In extreme cases, the reflectance interpolation and emittance uncertainties together can result in solar infrared temperature errors that exceed 10 and 1 K for ice and water-droplet clouds, respectively. For useful satellite angular configurations, the mean rms errors are less than 0.7 K.

Specification of the visible optical depth, viewing and illumination conditions, clear-sky temperature at each wavelength, cloud temperature, and particle size are the only parameters required to compute the radiances for the visible, solar infrared, infrared window, and split window in the absence of an atmosphere. When combined with the visible reflectance parameterization of Minnis et al. (1993b) and a representation of atmospheric gaseous absorption (e.g., Kratz 1995), the parameterization can be used to simulate the top of the atmosphere radiances for each wavelength. The limitations of the parameterizations developed here have been discussed and quantified in detail so that they may be applied confidently. Despite their shortcomings, these models are considerably more accurate and better understood than previous parameterizations. Because they are accurate and computationally economical, the models presented here should be valuable for simulating and retrieving cloud properties on a global scale.

## Acknowledgments

Fred Denn of AS&M, Inc., provided the simulated satellite viewing angle calculations and Bill Smith, Jr. of AS&M, Inc., prepared the satellite imagery. This research was supported by the Earth Observing System Interdisciplinary Program, NASA/Office of Mission to Planet Earth through the Clouds and Earth’s Radiant Energy System Project. Additional support was provided by the Department of Energy Environmental Sciences Division, Inter Agency Transfer Agreement ITF#21422116AQ through Pacific Northwest Laboratories and Inter Agency Agreement DE-AI02-97ER62341 as part of the Atmospheric Radiation Measurement Program.

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## APPENDIX

### Satellite Sampling and Node Interpolation Errors

#### Effect of satellite sampling on VIS reflectance interpolation errors

Figure A1 shows the frequency distributions of Θ observed from several satellites that are currently or will be used to measure cloud properties. Only scattering angles corresponding to *θ*_{0} < 82° and *θ* < 70° were considered in these plots because few satellite-observed reflectances are interpreted at greater zenith angles. A scattering angle was computed for each 0.25° of latitude and longitude viewed by the satellite. For the GOES and the morning-orbit Earth Observing System (EOS-AM) satellite, the observations were restricted to latitudes equatorward of 60°. GOES observations were simulated every half hour for both the equinox and solstice days to cover the extremes of the solar positions. The EOS-AM satellite is assumed to be in a Sun-synchronous orbit with a 1030 local time (LT) descending node. The calculations were performed for the boreal summer and winter solstices and the autumnal equinox. The Tropical Rainfall Monitoring Mission (TRMM) satellite is in a 35°-inclined orbit with a repeat cycle of 47 days. The usual NOAA satellite orbits were simulated with equatorial crossing times of 0730 (AM) and 1430 LT (PM) for the same days as the EOS satellite.

Figure A1a shows that scattering angles between 140° and 175° are observed most often by GOES while Θ < 60° is viewed infrequently. During the equinoxes, the most common scattering angle is ∼172°. This maximum shifts with solar declination to ∼149° at the solstices. Larger scattering angles are not observed during the solstices because the line of sight from the sun through the satellite is off the earth due to the height required for geostationary orbit. The maximum frequencies are near some of the angles with large errors (see Fig. 17) suggesting that the uncertainties in the modeled reflectance may be larger than the average given above. However, most of the angles with the largest reflectance errors are infrequently seen with the GOES geometry. A peak near 145° occurs for the EOS-AM (Fig. A1b) with a second maximum near Θ = 100°. TRMM (Fig. A1c) has the most Gaussian distribution of all with a peak between 105° and 130°, a relatively flat section of the error distribution for water droplets but not for CS. The NOAA-AM (Fig. A1d) has a very pronounced maximum near 60° with a weak secondary peak near 155°. A strong backscattering peak is missing because the satellite often views a darker area with *θ*_{0} > 82° at that hour. NOAA-PM (Fig. A1e) is more like EOS-AM but with maxima closer to 90° and 160°.

The mean and rms errors for each satellite configuration were recomputed by weighting each simulated observation by the appropriate frequency in Fig. A1. The results are summarized in Table A1. In most cases, the mean and rms errors are less than expected for no satellite-view weightings. Mean reflectances are greatest for the unweighted cases because the reflectances at small values of Θ are much larger than those at other angles. For water droplets, the reflectance uncertainties due to interpolation are smallest for the NOAA-AM orbits and greatest for the GOES and EOS-AM observations. The water-droplet errors and most of the satellite observations occur at Θ > 60° (see Fig. 17a). Therefore, the satellite and unweighted rms errors are not very different. The ice-model errors for the satellite viewing geometries are all significantly less than the unweighted errors due to the lack of observations at the smaller scattering angles. Because of fixed viewing geometry for a given area, the interpolation errors from GOES will probably not average out the bias for a given local time. The daily average error may approach zero, however, when all local hours are considered because of the wide range of Θ observed over the day. The other satellites view a particular area from a different angle each day, therefore, the mean error incurred at one scattering angle will likely be compensated to some extent by views from other angles on subsequent days. For example, during its 11-day repeat cycle, the AVHRR cross-track scanner views a particular area toward and away from the sun over a *θ*-range of ±70° giving a large range in Θ. While other satellite–solar configurations are expected to differ from those considered here (e.g., Fig. A1), it is apparent that views at Θ < 50° will be relatively infrequent. Thus, the sharp halo features in the VIS ice-crystal phase functions will rarely be a factor in the retrieval of cloud properties. However, the large reflectance errors expected for Θ > 174° will be encountered in the views from most satellites. Users of these models should be aware of the potential errors in these situations.

#### SI temperature errors due to interpolation

An independent dataset was created to determine the SI temperature errors due to the parameterizations and interpolations of emittance. The AD computations were performed for the T40 and 8-*μ*m models using 9 solar zenith, 9 viewing zenith, and 10 relative azimuth angles, 7 optical depths, 5 surface albedos ranging from 0 to 0.20, and with and without the solar component. The angles were selected to cover the full range of useful viewing and illumination conditions while differing from the lookup table nodes. One set of optical depths was selected to be approximately midway between the interpolation nodes and the other corresponds to the VIS model nodes. Five combinations of *T*_{c} and *T*_{s} were selected for the ice and liquid model calculations to cover a realistic range of values. For T40, *T*_{c}–*T*_{s} = 222 K–283 K, 203 K–309 K, 236 K–268 K, 264 K–283 K, and 257 K–309 K. The last three combinations were also used for the 8-*μ*m model in addition to *T*_{c}–*T*_{s} = 281 K–309 K and 287 K–293 K.

##### Nighttime errors

The off-node rms errors in the effective emittance parameterization for channel 3 in ε are 1.1% and 2.5% for the 8-*μ*m and T40 models, respectively, similar to the SEEs from the regression fits. Assuming that the results are typical of all of the models in this study, it can be concluded that the SEEs in Fig. 13 are representative of the parameterization errors for each of the six wavelengths.

For the T40 cases, the no-sun comparisons produced temperature errors of −0.1 ± 1.1 K for values of *τ* between the optical depth nodes. At the nodes, the errors are 0 ± 0.3 K. The mean optical depth interpolation error can be estimated using the rms mean which is 0 ± 1.0 K for the T40 model. For the 8-*μ*m data, the rms off- and on-node errors are 0.5 and 0.6 K, respectively. Thus, the Lagrangian interpolation introduces almost no additional error for the 8-*μ*m model. Most of the uncertainty arises for the thick, high cirrus cases where Δ*T*_{sc} is greatest. For *τ* < 8, the off-node T40 rms errors are only 0.3 K but 2.2 K for *τ* > 8.

To further illustrate the sensitivity to Δ*T*_{sc}, consider the case for the T40, 3.75-*μ*m model where *τ* = 12.2, the parameterized emittance error is 0.002, *T*_{s} = 309 K, and *T*_{c} = 203 K. This small emittance error (∼0.2%) translates to a temperature error of −10 K because the sensitivity factor is so large. Although the temperature errors at the optical depth nodes are only ∼1 K, the small interpolation error in emittance produces a large temperature error. Thus, the simulation of TOA SI temperatures for high-altitude cirrus over hot land surfaces can be subject to some significant uncertainties despite the small effective emittance errors. The IR and WS temperature errors are much smaller than the SI values because the magnitudes of Δ*R/R* in (13) are generally much less than the SI values.

In most cases, the SI temperature estimates from these parameterizations will be within ±0.5 K of the AD model results. Although the accuracy of this model exceeds that of previous parameterizations, especially for the SI spectra, there are some situations that will result in large temperature errors and, for remote sensing, large retrieval uncertainties. To minimize the effect of the sensitivity to emittance interpolation in high-contrast situations, it may be more accurate to interpolate using the TOA temperatures at the nodes. Using temperature instead of emittance as the dependent variable in the interpolation decreases the errors to 1.0 K for *τ* > 8 for the T40 case but does not decrease the errors for *τ* < 8 or for the 8-*μ*m model. Because it requires many fewer calculations, emittance interpolation is preferred. The largest errors could be reduced, however, if temperature interpolation were employed for larger ice-cloud optical depths only. For example, the overall T40 interpolation error is 0.6 K if temperature and emittance interpolations are used for *τ* > 8 and *τ* < 8, respectively.

##### Daytime errors

The uncertainties in the daytime SI temperatures may be greater than those at night because of the interpolation errors between the reflectance model nodes. The temperature errors were computed for the 8-*μ*m and T40 models during the daytime for a surface albedo of 0.05. Overall, there is no bias in the 8-*μ*m results with an rms error of 0.6 K, the same as the emittance-only errors. Thus, the Lagrangian interpolation yields negligible errors for the smooth 8-*μ*m SI phase function. The features in the ice-crystal phase function, however, appear to introduce some substantial uncertainty into the estimation of SI temperature during the daytime. Although 86% of the errors are smaller than ±1 K, nearly 2.5% of the simulated temperatures differ by more than 10 K from their AD counterparts.

The errors are plotted in Fig. A2 as means and standard deviations for each 5° of scattering angle. In Fig. A2a, it is clear that the greatest errors for the 8-*μ*m model are incurred for Θ > 155° with the maximum error of −1.0 ± 1.2 K. The T40 errors (Fig. A2b) are greatest in the backscattering direction, but they are much larger than the corresponding 8-*μ*m values. Significant errors also occur for T40 in the forward scattering directions for Θ < 50°. Between 50° and 140°, the interpolation errors are negligible. Thus, for most of the angles commonly observed by satellites (e.g., Fig. A1), the parameterized daytime SI temperatures will be within ±0.7 K of the AD values, especially for the water-droplet model. For the ice-crystal models, the user should take into account the potentially large errors in the SI cloud temperatures for Θ > 150°. Overall, the rms error for the T40 daytime results is ∼±1.0 K.

The SI temperature errors for the other simulated surface albedos are almost identical to those found for a surface albedo of 0.05. Thus, the parameterization represented by (11) accurately accounts for the underlying surface. The two models used in this error analysis are in the middle of the considered size distributions and the results should represent an average parameterization error for all of the particle sizes. Both greater and smaller errors will be encountered for a given particle size, however, depending on the variations in the scattering phase function.

Ice-crystal distributions developed for a new contrail (NCON), an older contrail (CON), and a large-particle cirrus (LPC) cloud.

Citation: Journal of the Atmospheric Sciences 55, 22; 10.1175/1520-0469(1998)055<3313:PORAEE>2.0.CO;2

Ice-crystal distributions developed for a new contrail (NCON), an older contrail (CON), and a large-particle cirrus (LPC) cloud.

Citation: Journal of the Atmospheric Sciences 55, 22; 10.1175/1520-0469(1998)055<3313:PORAEE>2.0.CO;2

Ice-crystal distributions developed for a new contrail (NCON), an older contrail (CON), and a large-particle cirrus (LPC) cloud.

Citation: Journal of the Atmospheric Sciences 55, 22; 10.1175/1520-0469(1998)055<3313:PORAEE>2.0.CO;2

Scattering phase function for three different ice cloud models (forward scattering maxima have been clipped) for (a) *λ* = 0.65 *μ*m and (b) *λ* = 3.75 *μ*m.

Scattering phase function for three different ice cloud models (forward scattering maxima have been clipped) for (a) *λ* = 0.65 *μ*m and (b) *λ* = 3.75 *μ*m.

Scattering phase function for three different ice cloud models (forward scattering maxima have been clipped) for (a) *λ* = 0.65 *μ*m and (b) *λ* = 3.75 *μ*m.

Theoretical Mie scattering phase functions for modified gamma distributions of water droplets at (a) *λ* = 0.65 *μ*m and (b) *λ* = 3.75 *μ*m.

Theoretical Mie scattering phase functions for modified gamma distributions of water droplets at (a) *λ* = 0.65 *μ*m and (b) *λ* = 3.75 *μ*m.

Theoretical Mie scattering phase functions for modified gamma distributions of water droplets at (a) *λ* = 0.65 *μ*m and (b) *λ* = 3.75 *μ*m.

Anisotropic reflectance factors for *λ* = 0.65 *μ*m, *μ*_{o} = 0.65, and *τ* = 1.0.

Anisotropic reflectance factors for *λ* = 0.65 *μ*m, *μ*_{o} = 0.65, and *τ* = 1.0.

Anisotropic reflectance factors for *λ* = 0.65 *μ*m, *μ*_{o} = 0.65, and *τ* = 1.0.

Same as Fig. 5 except for *λ* = 3.75 *μ*m.

Same as Fig. 5 except for *λ* = 3.75 *μ*m.

Same as Fig. 5 except for *λ* = 3.75 *μ*m.

Solar zenith angle dependence of cloud albedo at *λ* = 0.65 *μ*m.

Solar zenith angle dependence of cloud albedo at *λ* = 0.65 *μ*m.

Solar zenith angle dependence of cloud albedo at *λ* = 0.65 *μ*m.

Solar zenith angle dependence of cloud albedo at *λ* = 3.75 *μ*m.

Solar zenith angle dependence of cloud albedo at *λ* = 3.75 *μ*m.

Solar zenith angle dependence of cloud albedo at *λ* = 3.75 *μ*m.

Diffuse albedos for model clouds at *λ* = 0.65 *μ*m.

Diffuse albedos for model clouds at *λ* = 0.65 *μ*m.

Diffuse albedos for model clouds at *λ* = 0.65 *μ*m.

Diffuse albedo for model clouds at *λ* = 3.75 *μ*m. [Note scale differences between (a) and (b).]

Diffuse albedo for model clouds at *λ* = 3.75 *μ*m. [Note scale differences between (a) and (b).]

Diffuse albedo for model clouds at *λ* = 3.75 *μ*m. [Note scale differences between (a) and (b).]

Effective emittances for all calculations at *r*_{e} = 6 *μ*m and a regression fit of ε = 1 − exp[*a*(*τ*/*μ*)^{b}].

Effective emittances for all calculations at *r*_{e} = 6 *μ*m and a regression fit of ε = 1 − exp[*a*(*τ*/*μ*)^{b}].

Effective emittances for all calculations at *r*_{e} = 6 *μ*m and a regression fit of ε = 1 − exp[*a*(*τ*/*μ*)^{b}].

Variation of effective emittance with clear-cloud temperature difference from adding–doubling model calculations for *r*_{e} = 6 *μ*m.

Variation of effective emittance with clear-cloud temperature difference from adding–doubling model calculations for *r*_{e} = 6 *μ*m.

Variation of effective emittance with clear-cloud temperature difference from adding–doubling model calculations for *r*_{e} = 6 *μ*m.

Emittance parameterization errors. Here, Ch3′ refers to errors for emittances less than 1 only.

Emittance parameterization errors. Here, Ch3′ refers to errors for emittances less than 1 only.

Emittance parameterization errors. Here, Ch3′ refers to errors for emittances less than 1 only.

Brightness temperature differences from parameterizations for *T*_{s} = 295 K, *T*_{c} = 260 K, *τ* < 16, and *θ* = 30°.

Brightness temperature differences from parameterizations for *T*_{s} = 295 K, *T*_{c} = 260 K, *τ* < 16, and *θ* = 30°.

Brightness temperature differences from parameterizations for *T*_{s} = 295 K, *T*_{c} = 260 K, *τ* < 16, and *θ* = 30°.

Reflectance errors relative to AD calculations due to interpolation of model lookup tables. Thin lines indicate extreme errors; thick lines denote means at a given scattering angle.

Reflectance errors relative to AD calculations due to interpolation of model lookup tables. Thin lines indicate extreme errors; thick lines denote means at a given scattering angle.

Reflectance errors relative to AD calculations due to interpolation of model lookup tables. Thin lines indicate extreme errors; thick lines denote means at a given scattering angle.

Spherical albedo as a function of water path for model clouds. The average ice-crystal rms error is less than 3% for all of the satellite cases.

Spherical albedo as a function of water path for model clouds. The average ice-crystal rms error is less than 3% for all of the satellite cases.

Spherical albedo as a function of water path for model clouds. The average ice-crystal rms error is less than 3% for all of the satellite cases.

Cirrus clouds over the Coral Sea observed with *NOAA-11* AVHRR 4-km data at 0511 UTC 17 January 1993.

Cirrus clouds over the Coral Sea observed with *NOAA-11* AVHRR 4-km data at 0511 UTC 17 January 1993.

Cirrus clouds over the Coral Sea observed with *NOAA-11* AVHRR 4-km data at 0511 UTC 17 January 1993.

Same as Fig. 17 except for stratus clouds as indicated in bottom of Fig. 17a.

Same as Fig. 17 except for stratus clouds as indicated in bottom of Fig. 17a.

Same as Fig. 17 except for stratus clouds as indicated in bottom of Fig. 17a.

Multilevel cirrus and stratocumulus clouds east of Cape Hatteras, North Carolina, observed with *NOAA-14* AVHRR 1-km data at 1848 UTC 16 January 1997. In the lower panels, parameters for solid lines are A: *T*_{c} = 281 K, *r*_{e} = 8 *μ*m; B: *T*_{c} = 281 K, *r*_{e} = 12 *μ*m;C: *T*_{s} = 285 K, *T*_{c} = 256 K, *D*_{e} = 30 *μ*m; D: *T*_{s} = 281 K, *T*_{c} = 256 K, *D*_{e} = 30 *μ*m; for dashed line—*T*_{s} = 292 K, *T*_{c} = 256 K, *D*_{e} = 30 *μ*m.

Multilevel cirrus and stratocumulus clouds east of Cape Hatteras, North Carolina, observed with *NOAA-14* AVHRR 1-km data at 1848 UTC 16 January 1997. In the lower panels, parameters for solid lines are A: *T*_{c} = 281 K, *r*_{e} = 8 *μ*m; B: *T*_{c} = 281 K, *r*_{e} = 12 *μ*m;C: *T*_{s} = 285 K, *T*_{c} = 256 K, *D*_{e} = 30 *μ*m; D: *T*_{s} = 281 K, *T*_{c} = 256 K, *D*_{e} = 30 *μ*m; for dashed line—*T*_{s} = 292 K, *T*_{c} = 256 K, *D*_{e} = 30 *μ*m.

Multilevel cirrus and stratocumulus clouds east of Cape Hatteras, North Carolina, observed with *NOAA-14* AVHRR 1-km data at 1848 UTC 16 January 1997. In the lower panels, parameters for solid lines are A: *T*_{c} = 281 K, *r*_{e} = 8 *μ*m; B: *T*_{c} = 281 K, *r*_{e} = 12 *μ*m;C: *T*_{s} = 285 K, *T*_{c} = 256 K, *D*_{e} = 30 *μ*m; D: *T*_{s} = 281 K, *T*_{c} = 256 K, *D*_{e} = 30 *μ*m; for dashed line—*T*_{s} = 292 K, *T*_{c} = 256 K, *D*_{e} = 30 *μ*m.

Contrail cloudiness over the coast of Virginia observed with *NOAA-12* AVHRR 1-km data at 1214 UTC 26 September 1996.

Contrail cloudiness over the coast of Virginia observed with *NOAA-12* AVHRR 1-km data at 1214 UTC 26 September 1996.

Contrail cloudiness over the coast of Virginia observed with *NOAA-12* AVHRR 1-km data at 1214 UTC 26 September 1996.

Cirrus clouds over ARM Southern Great Plains central facility observed with *GOES-8* 4-km data at 0145 UTC 15 April 1995.

Cirrus clouds over ARM Southern Great Plains central facility observed with *GOES-8* 4-km data at 0145 UTC 15 April 1995.

Cirrus clouds over ARM Southern Great Plains central facility observed with *GOES-8* 4-km data at 0145 UTC 15 April 1995.

[dt19p,1p0g,12p0] Fig. A1. Scattering angles observed by various satellites for *θ*_{0} < 82°, *θ* < 70°.

[dt19p,1p0g,12p0] Fig. A1. Scattering angles observed by various satellites for *θ*_{0} < 82°, *θ* < 70°.

[dt19p,1p0g,12p0] Fig. A1. Scattering angles observed by various satellites for *θ*_{0} < 82°, *θ* < 70°.

Fig. A2. Scattering angle dependence of errors in 3.75-*μ*m equivalent blackbody temperatures at the top of the atmosphere relative to adding–doubling computations for two cloud models over surface with albedo of 0.05.

Fig. A2. Scattering angle dependence of errors in 3.75-*μ*m equivalent blackbody temperatures at the top of the atmosphere relative to adding–doubling computations for two cloud models over surface with albedo of 0.05.

Fig. A2. Scattering angle dependence of errors in 3.75-*μ*m equivalent blackbody temperatures at the top of the atmosphere relative to adding–doubling computations for two cloud models over surface with albedo of 0.05.

Refractive indices used for Mie and spheroid calculations.

Monodisperse ice cloud optical properties for GOES VIS and AVHRR channel 1 and for GOES channel 2 and AVHRRchannel 3.

Monodisperse ice cloud optical properties for GOES/AVHRR channels 4.

Monodisperse ice cloud optical properties for GOES/AVHRR channels 5.

Ice cloud model physical properties.

Asymmetry factors *g* for model particle size distributions.

Single-scattering albedos [si2]*ω̃*_{0} for model particle size distributions.

Extinction efficiency factors *Q* for model particle size distributions.

Reflectance errors due to interpolation between angle nodes.

Rms temperature differences between AD model and emittance parameterization.

Table A1. Reflectance interpolation errors convolved with the scattering angle frequencies in Fig. 15 for various satellite configurations.