## 1. Introduction

Models of ice-cloud bulk microphysical and single-scattering properties are essential to perform realistic radiative transfer calculations. Such calculations may be used, for example, in remote sensing applications for the inference of optical thickness and effective particle size, in numerical cloud-resolving models, or in general circulation models. The bulk properties include effective diameter *D*_{eff} or median mass diameter *D _{m}*, ice water content (IWC), extinction and scattering efficiencies

*Q*

_{ext}and

*Q*

_{sca}, respectively, extinction and scattering cross sections

*σ*

_{ext}and

*σ*

_{sca}, respectively, single-scattering albedo

*ω*, asymmetry parameter

*g*, and the scattering phase function

*P*(Θ), where Θ is scattering angle. Bulk properties are obtained by integration of individual ice-particle properties over both habit and size distributions.

Ice-cloud bulk single-scattering models assume some knowledge of the particle size and habit distributions as well as the individual particle single-scattering properties for each size and habit. The advent of new techniques for calculating individual particle single-scattering properties combined with a growing set of in situ measurements of ice clouds from various field campaigns provides an opportunity to develop bulk single-scattering models that are derived consistently for a wide range of instruments taking measurements from visible to far-infrared wavelengths.

Baum et al. (2005a, b) incorporated both ice-cloud in situ measurements and recent advances in scattering calculations for a variety of ice-particle habits with the goal of developing narrowband scattering models for use in remote sensing applications. Baum et al. (2005a) focused on the ice-cloud microphysical models, including the ice-particle size and habit distributions. Baum et al. (2005b) discussed the development of a set of narrowband visible and shortwave infrared models for the Moderate-Resolution Imaging Spectroradiometer (MODIS).

Our current work integrates the ice-particle size and habit distributions described by Baum et al. (2005a) with the database of scattering and absorption properties for ice plates, hollow columns, solid columns, 3D bullet rosettes, aggregates, and droxtals described by Yang et al. (2005) and Huang et al. (2004). This integration leads to the creation of a set of bulk single-scattering property models appropriate for use by high-resolution infrared spectrometers in the wavenumber spectrum between 100 and 3150 cm^{−1}, such as the Atmospheric Infrared Sounder (AIRS; Aumann et al. 2003) on the National Aeronautics and Space Administration (NASA) Earth Observing System (EOS) *Aqua* satellite. AIRS is a high-spectral-resolution IR sounder (*ν*/Δ*ν* = 1200, where *ν* is the wavenumber and Δ*ν* is the width of a band) with 2378 channels. AIRS measures radiances in the spectral region of 3.74–15.4 *μ*m. The AIRS data are used in conjunction with data from the Advanced Microwave Sounding Unit to provide the vertical profiles of atmospheric temperature and water vapor. However, the data from AIRS are also useful for inferring cloud properties such as cloud effective particle size *D*_{eff} and optical thickness *τ* with its high-spectral-resolution IR radiances (Wei et al. 2004; Li et al. 2005; Huang et al. 2004). One-dimensional variational and minimum-residual approaches are used to retrieve the *D*_{eff} and *τ* from the AIRS longwave-window- region (790–970 cm^{−1} or 10.31–12.66 *μ*m, and 1050–1130 cm^{−1} or 8.85–9.52 *μ*m) cloudy radiance measurements (Li et al. 2005).

The models discussed herein are built in a way that is consistent with those developed for the MODIS imager (Baum et al. 2005b) that have been implemented for the operational (“collection 5”) products. To compare properties inferred from AIRS and MODIS, which are both on the same EOS *Aqua* platform, it is preferable to use consistent bulk single-scattering models in their respective data analyses. In this third part of the series, we extend the method to provide spectral bulk single-scattering models that are appropriate for use with hyperspectral IR data.

Section 2 briefly describes the scattering and absorption property database and the microphysical data collected during various field campaigns. To create bulk single-scattering property models for ice clouds for a given wavenumber, the single-scattering properties for individual ice crystals must be integrated over particle size and habit distributions. Section 3 discusses the method used to integrate various properties over particle size and habit distributions. Results are provided in section 4, and section 5 contains conclusions.

## 2. Data and models

### a. In situ microphysical data

A full description of the in situ data is presented in Baum et al. (2005a) and will be briefly summarized herein. Particle size distributions (PSD) and other microphysical parameters such as median mass diameter *D _{m}*) and IWC have been developed for measurements in ice clouds in the midlatitudes, Tropics, and subtropics. These data are derived from the following five field campaigns: 1) the First International Satellite Cloud Climatology Project Regional Experiment (FIRE-I) in Madison, Wisconsin, in 1986; 2) FIRE-II held in Coffeyville, Kansas, in 1991; 3) the Atmospheric Radiation Measurement Program (ARM) intensive observation period (IOP) near Lamont, Oklahoma, in the spring of 2000; 4) the Tropical Rainfall Measuring Mission (TRMM) Kwajalein Experiment (KWAJEX) flights near Kwajalein, Marshall Islands, in 1999; and 5) the Cirrus Regional Study of Tropical Anvils and Cirrus Layers Florida-Area Cirrus Experiment (CRYSTAL-FACE) in 2002. Table 1 of Baum et al. (2005a) summarizes the pertinent information regarding the location, altitude, cloud temperature, sampling probes, and the minimum and maximum particle sizes sampled.

*D*

_{max}is the particle maximum dimension and

*n*(

*D*) is the particle concentration per unit volume. On a log–log plot of this equation,

*N*

_{0}is derived from the intercept,

*λ*is derived from the slope, and

*μ*is derived from the dispersion (Heymsfield et al. 2002). Note that this relationship reduces to an exponential distribution when

*μ*= 0. The values for the intercept, slope, and dispersion were derived for each PSD by matching the three moments (i.e., the first, second, and sixth moments) that provided the best fit over the measured particle size range (Heymsfield et al. 2002). The data are filtered by data from a Rosemount icing probe that detects even minute amounts of liquid water and also cloud temperature to ensure that the particle phase is ice. PSDs are used only from clouds that are colder than −25°C. Of the more than 4000 original PSDs, a total of 1117 PSDs remain after filtering by cloud temperature and the data from the Rosemount icing probe. Heymsfield et al. (2002, 2003, 2004, 2006 and Heymsfield and Miloshevich (2003) provide detailed discussion of the measurement techniques, data descriptions, and analyses and more comprehensive information on

*N*

_{0},

*λ*, and

*μ*.

### b. Ice-particle scattering and absorption properties

For IR radiative transfer calculations, the required single-scattering parameters include extinction and absorption efficiencies, extinction and absorption cross sections (and hence single-scattering albedo), and scattering phase function or asymmetry factor, which is the first-order moment of the phase function. In turn, the scattering and absorption properties of ice particles are determined by the habit (or shape) and size of the particles, the complex refractive index, and the incident wavelength. Figure 1 shows the refractive index for ice in the spectrum between 100 and 3250 cm^{−1} (Warren 1984). Absorption within the ice particles is proportional to the imaginary index of refraction. Strong regions of ice absorption are noted near 200, 800, 1600, 2200, and 3200 cm^{−1}.

Yang et al. (2005) discuss in detail the scattering and absorption properties for a set of ice-particle habits, with a brief overview provided herein. An extensive database of scattering and absorption properties has been developed for a set of ice particles that include droxtals (spheroidal particles with 20 facets), hexagonal plates, hollow columns, solid columns, three-dimensional bullet rosettes, and aggregates (Yang et al. 2005). The scattering and absorption properties of a particle are determined by the shape and size of the particle, the complex refractive index, and the incident wavelength. The database (Yang et al. 2005) contains properties for 49 wavenumbers between 100 and 3250 cm^{−1} (3 and 100 *μ*m), which are further interpolated to 3151 discrete wavenumbers on the basis of a third-order spline interpolation method. The 49 wavenumbers were chosen to account for the sharp gradients in the refractive index between 100 and 3250 cm^{−1} and to allow interpolation to a higher spectral resolution. For this work, the database has been interpolated to 1 cm^{−1} resolution, which is close to the advanced IR sounder spectral resolution. Therefore, at each wavenumber *ν*, there is a full set of infrared single-scattering properties for each of the ice crystal habits in 45 size bins spanning a range of 2–9500 *μ*m in terms of particle maximum dimension *D*. The database contains particle volume *V*(*ν,D*), projected area *A*(*ν,D*), asymmetry factor *g*(*ν,D*), scattering cross section *σ*_{sca}(*ν,D*), extinction cross section *σ*_{ext}(*ν,D*), and the scattering phase function *P*(Θ, *ν,D*) at 498 scattering angles Θ from 0° to 180°.

There is no single method that can provide scattering and absorption properties for nonspherical ice particles in the infrared spectral range. In an effort to provide continuous single-scattering properties based on the various numerical methods, Fu et al. (1998) and Yang et al. (2005) discuss in detail the use of a compositing approach; a brief summary is presented here.

The finite-difference time domain (FDTD) method (Yee 1966; Taflove 1995; Yang and Liou 1996a; Sun et al. 1999, 2002) is used when the size parameter is less than 20. The size parameter for an ice particle is defined as *πD*/*λ*, where *D* is the maximum dimension of the particle and *λ* is the incident wavelength. For size parameters larger than 20, the improved geometric optics method (IGOM; Yang and Liou 1996b) is used. However, a discontinuity exists in the infrared spectrum between the two solutions at size parameters ranging from approximately 20 to 60. Some studies (Fu et al. 1998, 1999; Baran et al. 2003) attribute this discontinuity to a tunneling effect that is neglected in the ray-tracing approach. The composite method seeks to mitigate the problems in the region of discontinuity. After Fu et al. (1998, 1999), the present implementation of the compositing approach combines results from the FDTD method, the IGOM, and Lorenz–Mie theory. The composite method is based on a linear combination of the equivalent-spherical solution and the IGOM results for moderate to large size parameters. The weighting coefficients in the combination are selected to achieve a smooth transition from the FDTD solution to the composite solution. The Lorenz–Mie solutions for spheres with equivalent volumes are applied to hollow columns and plates; the Lorenz–Mie solutions for spheres with equivalent volume-to-projected-area ratios are applied to other habits.

The previous compositing approach is used for extinction efficiency, absorption efficiency, and asymmetry factor for size parameters between 20 and 60. For the scattering phase function, however, the composite method is unnecessary. The relative angular distribution of the scattered energy (i.e., the normalized phase function) computed from IGOM is influenced by the strong absorption within the ice particles in the IR. Because of the strong absorption within the particles, the phase function is basically featureless. The phase functions computed from the IGOM are considered to be approximate solutions for particles having large size parameters (Lee et al. 2003). The scattering phase functions generated by the FDTD and IGOM methods are so similar for the intermediate size parameters (i.e., around 20) that no compositing method is required. For size parameters ≤20, the FDTD-generated phase functions are used; the IGOM-generated phase functions are used for size parameters >20.

## 3. Bulk microphysical and optical properties

The discussion in this section pertains to the derivation of bulk microphysical and scattering/absorption properties that are obtained by integrating the single-scattering properties for the simulated ice particles (section 2b) over each of the particle size distributions (section 2a) and habit distributions (discussed below in section 3c).

### a. Microphysical properties

*f*(

_{h}*D*) is the ice-particle habit fraction for habit

*h*,

*D*

_{min}and

*D*

_{max}describe the minimum and maximum particle sizes in the distribution,

*n*(

*h,D*) is the number distribution of a specific particle habit

*h*for size

*D*, and

*A*(

_{h}*D*) and

*V*(

_{h}*D*) are the area and volume of a specific particle of habit

*h*for size

*D*, respectively. The total mass is obtained by multiplying the total volume by the bulk ice density (0.917 g cm

^{−3}).

*D*

_{eff}, which is proportional to the ratio of the total volume to the total projected area for a given particle size distribution. After Foot (1988), Francis et al. (1994), and King et al. (2004),

*D*

_{eff}is defined as

### b. Spectral bulk scattering properties

This section provides the derivation of spectral bulk single-scattering properties following the approach discussed in Baum et al. (2005b). The only difference in method is that these single-scattering properties are derived at an individual wavenumber rather than integrated over a spectral response function.

*ν*is given by

*M*is the number of habits,

*D*is particle size, and

*n*(

*D*) is the particle density. The habit fraction is defined so that, for each size bin,

*ω*

*(Θ) is given by*P

*is also weighted by the scattering cross section:*g

### c. Particle habit distribution

Baum et al. (2005a) developed the following approach to develop a habit mixture that matched IWC and *D _{m}* more closely with the in situ data than did the assumption of a single habit in the PSDs. Although the number of small particles is still uncertain with each PSD, results are shown later that explore the sensitivity of the scattering properties to the number of small particles. Each PSD was divided into four domains based on particle maximum dimension:

*D*

_{max}≤ 60

*μ*m, 60 <

*D*

_{max}≤ 1000

*μ*m, 1000 <

*D*

_{ma}

*≤ 2000*

_{x}*μ*m, and

*D*

_{max}> 2000

*μ*m. The following restrictions were invoked in developing the habit mixture. Droxtals were used to represent only the smallest particles, that is, those with

*D*

_{max}less than 60

*μ*m. Plates were used for intermediate-sized particles no larger than 1000

*μ*m. Both solid and hollow columns are employed for particles no larger than 2000

*μ*m, based on earlier observations that neither plates nor columns tend to occur at sizes much larger than 1500–2000

*μ*m (Ono 1969; Auer and Veal 1970). Aggregates were used only for particles larger than 1000

*μ*m. Based on these restrictions and size domains, the following distribution was suggested:

*D*

_{max}≤ 60

*μ*m, 100% droxtals; 60 <

*D*

_{max}≤ 1000

*μ*m, 15% 3D bullet rosettes, 50% solid columns, 35% plates; 1000 <

*D*

_{max}≤ 2000

*μ*m, 45% hollow columns, 45% solid columns, 10% aggregates; and

*D*

_{max}> 2000

*μ*m, 97% 3D bullet rosettes, 3% aggregates. As noted in Baum et al. (2005a), this habit distribution is one of many that could be derived to minimize differences between the in situ and derived median mass diameter and ice water content values for the set of PSDs.

## 4. Results

In this section we present the bulk scattering properties obtained from a set of 1117 particle size distributions discussed in detail in Baum et al. (2005a). Results are presented as functions of both the effective diameter and wavenumber.

Figure 2 shows the values for * g* and

*ω*

*ν*= 943 and 560 cm

^{−1}. At 943 cm

^{−1}, particle absorption is very strong; however, particle absorption is weak at 560 cm

^{−1}. The individual values for

*ω*

*(Fig. 2b) are calculated for each PSD at*g

*ν*= 943 cm

^{−1}and are color coded by field campaign. Figures 2c and 2d show

*ω*

*, respectively, at*g

*ν*= 560 cm

^{−1}. The

*D*

_{eff}values range from 40 to 250

*μ*m. The smallest

*D*

_{eff}values correspond to the flight track near Nicaragua from a single day of CRYSTAL-FACE PSDs, which were obtained from the coldest cloud in the dataset and which have some of the narrowest size distributions and lowest IWC values. As a way to gain some sense of the sensitivity of the bulk scattering parameters to small particles, the CRYSTAL-FACE PSDs are modified following Baum et al. (2005b). That is, the particle number densities in the smallest size bins (<20

*μ*m) are multiplied by 100 in the first set of calculations and by 1000 in the second set. This is an arbitrary enhancement of only the smallest particles that is performed solely to gain some insight as to the sensitivity of the bulk single-scattering parameters to small particles. Figure 2 also shows the results of multiplying the smallest particles in the CRYSTAL-FACE PSDs by factors of 100 (color coded in magenta) and 1000 (color coded in orange).

Note that at 943 cm^{−1} *ω**D*_{eff} decreases (Fig. 2a), whereas the opposite behavior is seen for *ω*^{−1} (Fig. 2c). If one interprets the single-scattering albedo as the probability that a photon will be scattered within the ice particle given an extinction event (Thomas and Stamnes 1999), then the probability of scattering will be lower for small particles at a highly absorbing wavenumber (943 cm^{−1}) than at a less absorbing wavenumber (560 cm^{−1}).

As shown in both Figs. 2b and 2d, there is a tendency for * g* to decrease with

*D*

_{eff}, which is attributed to the influence of small particles (e.g., Garrett et al. 2003). Note that

*decreases quickly as the number of small particles increases. This is expected because small particles tend to scatter isotropically.*g

As in Baum et al. (2005b), our next goal is to develop single-scattering properties at each wavenumber for a reduced set of *D*_{eff} values to simplify their application in radiative transfer modeling. In this case, 18 discrete sets of single-scattering properties are developed for *D*_{eff} = 10–180 *μ*m in increments of 10 *μ*m. We noted previously that the way we chose to obtain the bulk scattering parameters for *D*_{eff} at sizes of less than ∼40 *μ*m was to modify the PSDs by increasing the number of small particles (<20 *μ*m) relative to the number of larger particles. The PSDs that fall within a narrow region around each target *D*_{eff} values are averaged; the width of the region is chosen such that between 10 and 20 individual PSDs can be averaged for each *D*_{eff}. For each *D*_{eff} value, a mean and standard deviation are calculated for each microphysical and single-scattering parameter.

Figure 3 presents five different bulk scattering parameters over the wavenumber range from 100 to 3250 cm^{−1}: extinction cross section (cm^{2}; Fig. 3a), absorption cross section (cm^{2}; Fig. 3b), extinction efficiency (Fig. 3c), absorption efficiency (Fig. 3d), and asymmetry factor (Fig. 3e). Results are shown for five *D*_{eff} models ranging from 20 to 100 *μ*m. It is evident from Fig. 3 that the single-scattering properties are dependent on the *D*_{eff}, with the sensitivity increasing as *D*_{eff} decreases. Peaks in the absorption cross section and absorption efficiency are noted in the regions of strong absorption (i.e., higher values in the ice imaginary index of refraction). It is interesting to note the sensitivity of asymmetry factor and single-scattering albedo to small particles for the 20-*μ*m *D*_{eff} model results that is predominant across the spectrum. Small particles consist solely of droxtals, and the scattering properties of these particles are unique (Yang et al. 2003).

Figure 3 shows the bulk scattering properties over the entire wavenumber range from 100 to 3250 cm^{−1}, but radiative transfer calculations must also account for atmospheric absorption by such constituents as water vapor and ozone. The impact of ice clouds on top-of-atmosphere radiances will be greatest in the window regions of the atmosphere, that is, in regions that are relatively transparent to trace gases, carbon dioxide, ozone, and water vapor. There are several primary window regions between 100 and 3250 cm^{−1}. The spectral region from 400 to 590 cm^{−1} may be thought of as a “dirty” window because it can be moderately transparent for cold, dry conditions such as may be found in polar regions. Turner et al. (2003) discuss the potential for using data within this window to aid in determination of cloud thermodynamic phase. The classic thermal IR window broadly spans from 833 to 1250 cm^{−1} (i.e., from 8 to 12 *μ*m). The 800–980 cm^{−1} portion of this broad window band encompasses the “split window” region familiar to users of spaceborne imagers that have 11- and 12-*μ*m bands. The 1100–1250 cm^{−1} region encompasses a less transparent portion of the thermal IR window. Last, the 2400–2900 cm^{−1} window covers a relatively clear spectral range.

In the range from 400 to 590 cm^{−1}, there is some sensitivity of the various single-scattering properties to small particles that are predominant in the 20-*μ*m *D*_{eff} model. As discussed earlier, this model is derived from artificially increasing the number of particles in the CRYSTAL PSDs for sizes less than 20 *μ*m, that is, in the seven smallest particle size bins. At these small particle sizes, the habit consists solely of droxtals, and these particles have unique single-scattering properties.

The extinction and absorption cross sections in the 800–1200 cm^{−1} spectral region tend to increase monotonically with *D*_{eff} but display little sensitivity to wavenumber. The extinction efficiency has a value near 2 for the larger *D*_{eff} values. However, the extinction efficiency displays the most sensitivity to wavenumber for the 20-*μ*m model and reaches a minimum value at approximately 930 cm^{−1}. The asymmetry factor for *D*_{eff} > 20 *μ*m has values above 0.9 but decreases for the 20-*μ*m model. As particle size becomes much smaller than the wavelength, the scattering tends to become more isotropic. If the scattering is completely isotropic, the asymmetry factor would be zero. The decrease in asymmetry factor with *D*_{eff} is expected given the greater influence of the smallest particles in the size distribution. The single-scattering albedo tends toward an asymptotic value of 0.5 for the larger *D*_{eff} values but is generally lower for the 20-*μ*m *D*_{eff} model.

Figure 4 shows the scattering phase function as a function of *D*_{eff} at two wavenumbers: 560 and 943 cm^{−1}. Both sets of scattering phase functions are similar in that the magnitude of the forward peaks in the scattering phase functions increases with *D*_{eff}. This is expected because smaller particles tend to scatter more isotropically. The phase functions are basically smooth except for the case in which droxtals exert a strong influence for the 20-*μ*m *D*_{eff} model.

The variation of the bulk single-scattering properties with respect to *D*_{eff} is smooth enough to allow interpolation to a higher *D*_{eff} resolution. It is possible to interpolate the bulk single-scattering properties to 1-*μ*m resolution in *D*_{eff}, creating grids with 1-cm^{−1} and 1-*μ*m resolution. In Fig. 5, contour plots of extinction efficiency _{ext} (Fig. 5a), scattering efficiency _{sca} (Fig. 5b), * g* (Fig. 5c), and

*ω*

*D*

_{eff}< 50

*μ*m. There are some interesting observations to make regarding these results. First, note the sharp changes from absorption to scattering regimes, for example, at 1000 and 700 cm

^{−1}. Also,

Q

_{sca}is noted to decrease with increasing

*D*

_{eff}across most of the IR region of interest.

## 5. Summary

This study reports on the development of bulk scattering models for ice clouds that are appropriate for use in hyperspectral radiative transfer simulations at 1-cm^{−1} spectral resolution from 100 to 3250 cm^{−1}. The models incorporate a new database of individual ice-particle scattering properties that includes faceted small particles called droxtals, 3D bullet rosettes, hexagonal solid and hollow columns, aggregates, and hexagonal plates. The database provides scattering properties for each habit in 45 size bins, ranging from 2 to 9500 *μ*m, at a spectral resolution of 1 cm^{−1} that was interpolated from 49 wavelengths for which the light-scattering properties were computed. Bulk models are developed at each wavenumber by integrating each property (e.g., scattering phase function, extinction and absorption cross sections, single-scattering albedo, or asymmetry factor) over both particle habit and size distributions. A set of 1117 PSDs is used in the analyses, each of which was obtained at a cloud temperature lower than −25°C. This set is taken from a larger set of more than 4000 PSDs, which were developed from analysis of FIRE-I, FIRE-II, the ARM (2000) IOP, TRMM-KWAJEX, and CRYSTAL-FACE data. The ensuing set of models at each wavenumber is simplified into a set of 18 *D*_{eff} values ranging from 10 to 180 *μ*m to simplify the process of performing radiative transfer simulations. Each *D*_{eff} model includes microphysical and scattering properties such as median mass diameter, ice water content, single-scatter albedo, asymmetry factor, extinction efficiency, and scattering phase function. The spectral models are appropriate for use by infrared spectrometers such as EOS AIRS and the National Polar-Orbiting Environmental Satellite System (NPOESS) Cross-Track Infrared Sounder (CrIS).

## Acknowledgments

This research was sponsored by the Earth Science Enterprise. The authors specifically acknowledge the support and encouragement of Dr. Hal Maring of the NASA Radiation Program at NASA headquarters. Ping Yang also acknowledges support from the National Science Foundation (ATM-0239605) and the NASA Radiation Sciences Program (NNG04GL24G). The views, opinions, and findings contained in this report are those of the authors and should not be construed as an official National Oceanic and Atmospheric Administration or U.S. government position, policy, or decision.

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+ The National Center for Atmospheric Research is sponsored by the National Science Foundation.