Notice: Wiley Online Library will be unavailable on Saturday 27th February from 09:00-14:00 GMT / 04:00-09:00 EST / 17:00-22:00 SGT for essential maintenance. Apologies for the inconvenience.
 This study investigates the relationship between ice water content (IWC) and equivalent radar reflectivity (Ze) at 94 GHz for clouds consisting of nonspherical ice particles with geometrical shapes of hexagonal solid and hollow columns, plates, 6-branch bullet rosettes, aggregates, and droxtals. The IWC is calculated from a set of 1119 ice particle size distributions (PSDs) measured during several field campaigns, which are discretized to 46 size bins based on particle maximum dimensions ranging from 2 to 10500 μm. The Ze at 94 GHz is calculated from the radar backscattering properties obtained by integrating over the PSD and chosen particle habit distributions. The influence of ice habit on the Ze-IWC relationship is investigated for ice clouds composed of individual ice particle habits and a habit mixture. The Ze-IWC relationship is found to be sensitive to cloud effective particle size and cloud temperature. For an ice cloud with a given IWC, the Ze tends to increase with increasing effective particle size. Similarly, the Ze generally increases with increasing cloud temperature, at least for clouds with IWC over 0.01 g/m3. These features are consistent with the observed relationship between effective particle sizes and cloud temperatures. The present investigation of the effect of temperature on the Ze-IWC relationship indicates that including temperature in the Ze-IWC relationship may not improve the estimates of IWC. However, the dependence of the Ze-IWC relationship on the effective particle size within a given temperature range is more pronounced, and may be potentially useful for inferring the cloud effective particle size from the Ze-IWC relationship.
If you can't find a tool you're looking for, please click the link at the top of the page to "Go to old article view". Alternatively, view our Knowledge Base articles for additional help. Your feedback is important to us, so please let us know if you have comments or ideas for improvement.
 Clouds generally cover between 65-70% of the Earth. Approximately, 30% of these clouds reside at heights corresponding to pressures lower than 400 hPa [e.g., Wylie et al., 2005; Hong et al., 2007]. These high-altitude ice clouds are composed of nonspherical particles. Synoptic cirrus, formed in environments of relatively low updraft velocities, and tend to be composed of pristine habits as droxtals, hexagonal columns and plates, bullet rosettes, and aggregates of these habits. However, in convective situations the habits of ice particles tend to be much more complex.
 In the past decade, significant efforts have been focused on the calculation of the scattering and absorption properties of these ice particles [e.g., Macke et al., 1998; Mishchenko et al., 2000; Bailey and Hallet, 2004; Heymsfield and Miloshevich, 2003; Yang et al., 2005; Baum et al., 2005a]. Recent improvements offer the capabilities to infer the scattering properties consistently over the electromagnetic spectrum from the ultraviolet (UV) through the far infrared (Far IR). However, relatively little research has focused on the interpretation of millimeter wavelength radar measurements of ice clouds based on the calculated scattering/absorption properties of nonspherical particles.
 This work is aimed at understanding the effect of ice particle nonsphericity on the relationship between ice water content (IWC) and equivalent radar reflectivity (Ze). In particular, the focus is on measurements offered by CloudSat, a spaceborne radar launched on 28 April 2006, which provides millimeter wavelength measurements at 94 GHz [Stephens et al., 2002].
 Additional, the relationship between cloud temperature and particle size for ice clouds has been investigated [e.g., Heymsfield and Platt, 1984; Garrett et al., 2003]. There is some evidence that the Ze-IWC relationship is sensitive to cloud temperature [e.g., Sassen et al., 2002; Boudala et al., 2006]. Boudala et al.  developed an IWC retrieval algorithms based on temperature and Ze using ice particle distributions measured in stratiform ice clouds in midlatitude and Arctic regions and assumed irregular ice particle shapes represented by aggregates of plates and dendrites.
 In this paper we explore the sensitivity of a derived Ze-IWC relationship to assumed ice particle habit. The basis for this analysis is a set of 1119 ice PSDs measured during several field campaigns in tropical and midlatitude regions, which are described in detail by Baum et al. [2005a]. The sensitivity of Ze-IWC relationships to nonspherical ice particle habits is investigated on the basis of a set of six habits (hexagonal solid and hollow columns, plates, 6-branch bullet rosettes, aggregates of columns, and droxtals). These are the same habits as those used for the bulk scattering models from visible through the Far-IR wavelengths in some previous studies [e.g., Platnick et al., 2003; King et al., 2004, 2006; Yang et al., 2005; Baum et al., 2005a, 2005b, 2007]. The effect of cloud environment temperature and ice particle size on the Ze-IWC relationship is also investigated.
2. Data and Methodology
Sassen et al.  introduced three approaches to derive the Ze-IWC relationship. In this study, we employ an algorithm to derive Ze from ground-based or airborne microphysical measurements. A set of PSDs used in this study were obtained from in situ measurements in several field campaigns covering tropical to midlatitude regions. The tropical measurements used in this study include two campaigns conducted in Kwajalein, Marshall Islands in 1999 under the auspices of the Tropical Rainfall Measuring Mission (TRMM) [Stith et al., 2002, 2004], and the Cirrus Regional Study of Tropical Anvils and Cirrus Layers (CRYSTAL) Florida Area Cirrus Experiment (FACE) in 2002. The midlatitude measurements include the First International Satellite Cloud Climatology Project Regional Experiments (henceforth FIRE-1) in Madison, Wisconsin in 1986, Coffeyville, Kansas in 1991 (FIRE-II), and the Atmospheric Radiative Measurement Program (ARM) Intensive Operational Period (IOP) near Lamont, Oklahoma in 2000. Detailed information about the microphysical measurements are provided by Miloshevish and Heymsfield , Heymsfield et al. [2002, 2003, 2004], and Heymsfield and Miloshevich . A resulting set of 1119 PSDs are summarized by Baum et al. [2005a]. Each PSD is represented in the form of a gamma distribution [e.g., Kosarev and Mazin, 1991; Mitchell, 1991; Heymsfield et al., 2002] as follows:
where D is the maximum dimension of an ice crystal particle, N(D) is the number density of ice crystal particles with a D, N0 is the intercept, λ is the slope, and μ is the dispersion.
 The IWC is derived from
where ρ is the ice density with a value of 0.917 g cm−3, fi(D) = 1, where i denotes the ice crystal habit in the ice cloud, fi(D) is the ice particle habit fraction for habit i at a D, Vi(D) is the volume of the habit i for a given D, and Dmin and Dmax are the minimum and maximum sizes of D in the given particle size distribution N(D), respectively.
 The IWC for a cloud composed of either a single habit (i = 1) or a given habit mixture (i > 1) is calculated from equation (2) for each of the 1119 PSDs. Different ice cloud habit distributions have been used for ice cloud retrievals from solar and infrared measurements [e.g., Yang et al., 2005; Baum et al., 2005b; King et al., 2004, 2006; Hong, 2007a, 2007b]. The habit distribution derived by Baum et al. [2005a] for MODIS Collection 5 cloud retrieval [King et al., 2006] is used in this study. The habit distribution consists of 100% droxtals when D < 60 μm, 15% bullet rosettes, 50% solid columns, and 35% plates when 60 μm < D < 1000 μm, 45% hollow columns, 45% solid columns, and 10% aggregates when 1000 μm < D < 2500 μm, and 97% bullet rosettes and 3% aggregates when D > 2500 μm.
 We assume that the Ze-IWC relationship has a form of IWC = aZeb, where IWC is in units of g m−3 and Ze is in units of mm6 m−3 (dBZ in terms of 10log Ze). The radar equivalent reflectivity factor Ze at horizontal (vertical) copolarization in units of mm6 m−3 is defined as [e.g., Atlas et al., 1995; Donovan et al., 2004; Sato and Okamoto, 2006; Hong, 2007a]
where λ is the wavelength at 94 GHz, σi is the backscattering cross section for the ith ice crystal habit at a D. The nonspherical ice particles in general have been assumed to be randomly orientated so that Shh = Sw and Shv = Svh. The backscattering cross section σ for each of the habits is computed from the DDA model [Hong, 2007a] at 46 discrete values of D in a range of 2–10500 μm.
where Ai(D) is the averaged projected area of the habit i for a given D.
Figure 1 shows the Ze-IWC relationship for clouds composed of six individual habits: hexagonal solid and hollow columns, plates, 3D bullet rosettes, aggregates, and droxtals. The differences in the Ze-IWC relationships for different habits show some sensitivity to the choice of habit for deriving the relationship.
 On the basis of the mass-volume-size relationship assumed for each of the 6 individual habits [Yang et al., 2005; Hong, 2007a, 2007b], a value of IWC can be calculated for each of the PSDs (i.e., equation (2)). These IWC values can be compared to those derived using the Ze-IWC relationships for the various habits shown in Figure 1, with results shown in Figure 2. The correlation coefficients of the IWC values are lower for hollow and solid columns and 3D bullet rosettes than for plates, droxtals, and aggregates. The correlation coefficient for aggregates is the highest of the various individual habits. This is in agreement with the representation of aggregates for irregular ice particles by Boudala et al. .
 Under natural conditions, ice clouds consist of a variety of habits, with the smallest particles having aspect ratios of near unity (like droxtals) and larger particles with various shapes. It may be unrealistic to apply the Ze-IWC relationships shown in Figure 1 to naturally occurring ice clouds. To gain some sense of the variability caused by the assumption of habit, however, we can develop a Ze-IWC relationship from the entire set of PSDs based on this set of six individual habits (i.e., 6 × 1119 pairs of IWC and Ze) to build the Ze-IWC relationship. The resulting Ze-IWC relationship is shown in Figure 3 along with those previously shown in Figure 1. It is clear that the Ze-IWC relationships for ice clouds composed of individual habits have distinct differences. The relationship is very similar for ice clouds composed of solid columns, hollow columns, and 3D bullet rosettes. With a given value of Ze, the inferred IWC can vary by a factor of 1.5–2.0. In particular, the variability in IWC increases when Ze has negative values of dBZ. When the Ze values are above 0 dBZ, the Ze-IWC relationship more closely approximates the individual relationships for aggregates, droxtals, and plates. However, when the Ze values are less than 0 dBZ, the Ze-IWC relationship more closely approximates the individual relationships for 3D bullet rosettes, solid columns, and hollow columns.
 In the present study, the effect of the particle effective size (De) on the Ze-IWC relationship is investigated, with results shown in Figure 4. Instead of using individual habits, a habit mixture based on the study by Baum et al. [2005a] is assumed, which was derived by comparing the calculated median mass equivalent diameters and IWC from in situ measured PSD with those in situ measurements. For each of the 1119 PSDs, the De is calculated from equation (4). The 1119 values of De range in value from less than 50 μm to greater than 200 μm. Six groups are formed with De ranging from 50 μm to 200 μm at an interval of 25 μm. Two additional groups are formed with De < 50 μm and De > 200 μm. The coefficient a and exponent b for the Ze-IWC relationships are given in Table 1 for the 8 groups of De.
Table 1. Fitting Coefficient a and Exponent b for the Relationships Between Ice Water Content (IWC) and Equivalent Radar Backscattering Reflectivity (Ze) at 94 GHz for Clouds Consisting of a Mixture of Ice Particle Habitsa
Ice Cloud Properties
The results are provided for a number of ranges of effective particle sizes (De) and cloud temperatures (T).
Effective particle size, De (μm)
De < 50
50 < De < 75
75 < De < 100
100 < De < 125
125 <De < 150
150 < De < 175
175 < De < 200
200 < De
Temperature, T (°C)
−30 < T < −25
−35 < T < −30
−40 < T < −35
−45 < T < −40
−50 < T < −45
T < −50
 The sensitivity of the Ze-IWC relationship to De is shown in Figure 4a. In general, for a given Ze, the IWC increases with decreasing De. In contrast, for a given IWC, Ze increases with increasing De. The slopes of the Ze-IWC relationships are close for Dm > 50 μm but the slope for the smallest value of De is different. The smallest values of De are mostly observed in the CRYSTAL-FACE (Figure 5). The distinct different slope of the Ze-IWC relationship for these ice clouds reveals again the influence of nonsphericity of ice particles on the Ze-IWC relationship. Chepfer et al.  found that the main habits of the ice particles observed in the CRYSTAL-FACE are hexagonal columns. However, the habit mixture derived by Baum et al. [2005a] is used to derive the Ze-IWC relationship in the present study.
 Moreover, the Ze-IWC relationships for De in the range of 50-100 μm and for De > 100 μm have similar slopes. These features are indicated by the values of the coefficient a and exponent b shown in Table 1. The regular dependence of the Ze-IWC relationships on De, except for the smallest De, may be potentially useful for deriving the De from observed Ze for a given IWC or to derive the IWC from the observed Ze for a given De. This result indicates that the vertical distributions of De or IWC could be derived from similar lookup table as Figure 4a.
Liu and Illingworth  and Sassen et al.  documented that the inclusion of temperature for retrieving IWC from Ze can improve the accuracy of retrieved IWC. Recently, Boudala et al.  developed a parameterized radar retrieval algorithm of IWC in terms of temperature and Ze which is based on in situ aircraft measurements. Since cloud temperature is given for each of our PSDs, the dependence of the Ze-IWC relationship on temperature is investigated. Figure 4b shows the Ze-IWC relationships for six groups of temperatures. The coefficient a and exponent b of the Ze-IWC relationships for the six groups are also listed in Table 1.
 Unlike the systematic effect of De on the Ze-IWC relationship, the effect of temperature on the Ze-IWC relationship shows more variability. While Boudala et al.  suggested that Ze generally increases with increasing temperature for a given IWC; in this study, this feature is generally observed only when Ze are above −10 dBZ. Thus one cannot draw firm conclusions from the current analysis that an explicit inclusion of temperature in the Ze-IWC relationship can improve the accuracy of IWC derived from Ze.
 The effect of temperatures on De has been investigated by numerous groups [e.g., Ou and Liou, 1995; Ou et al., 1995; Wyser, 1998; Garrett et al., 2003]. If De should be a function of temperature, it would make sense to include temperature in the Ze-IWC relationship. The De as a function of temperature for the 1119 measurements during the CRYSTAL-FACE, TRMM, ARM, FIRE-I, are FIRE-II are shown in Figure 5. In general, the De of ice clouds increase with increasing temperatures. However, the relationship between De and temperature shows much variability. This feature is distinctly shown by the evident separation of the measurements in the TRMM campaign. The TRMM measurements came from cirrus anvils, and thus from an environment denoted by high updraft velocities, whereas the other PSDs came from cirrus having much lower updraft velocities. Our analysis suggests that the temperature cannot be included into the Ze-IWC relationship through a common relationship between temperature and De.
 Because of the pronounced variability in the relationship between De and temperatures, the effect of De on the Ze-IWC relationship for ice clouds is investigated for two temperature ranges with different De ranges. Note that the deriving Ze-IWC relationships do not involving the temperatures directly. Two temperature ranges of −50°C to −40°C and −40°C to −25°C are used to separate the measured ice cloud PSDs first. The separated PSDs are then used to derive the Ze-IWC relationships for the De in the range of 50–150 μm and 50–200 μm at the two temperature ranges, respectively. The two temperature ranges and De ranges are chosen in order to have sufficient samples for the analyses. Similarly to the results shown in Figure 4a, the De values are grouped with an interval of 25 μm. The Ze-IWC relationships in Figure 6 show a similar feature as Figure 4a, but for a given temperature range, the dependence of Ze-IWC relationship on De is more pronounced.
 For the derived Ze-IWC relationships in the two given temperature ranges (Figure 6), the IWC values are compared to those from the Ze-IWC relationships without considering the influence from the temperatures (Figure 4a) for different De. The correlations between the two derived IWC are similar, and the average deviations of the two derived IWC with respect to the IWC calculated from the particle size distributions are similar. This indicates again that including temperature for the Ze-IWC relationship does not provide a significant improvement of the accuracy of the IWC from the Ze-IWC that includes cloud temperature.
 However, Figure 6 also indicates that separating the effects of De and temperatures of ice clouds on the Ze-IWC relationships may be useful for inferring the De from the Ze-IWC relationship. For different De, the exponent b of the Ze-IWC relationships are similar. This agrees well with the results presented by Brown et al. , who showed that the exponents of the Ze-IWC relationships for inverse-exponential size distributions of varying scale diameter are the same. Thus the mean values of the exponents for the Ze-IWC relationships for different De ranges, with a size interval of 25 μm are used for the exponent a of the Ze-IWC relationship developed for the entire size range of 50–200 μm. The coefficients a of the Ze-IWC as a function of De based on the mean values of each size bin are shown in Figure 7. A fitting is performed for the relationships between De and the coefficient a in the range of 50 < De < 200 μm.
 The Ze-IWC relationships for different De, developed for two temperature ranges of −50°C < T < −40°C and −40°C < T < −25°C, are shown in Figure 8. The IWC and Ze calculated from the individual PSDs are also shown in the figure. The relationships among the Ze, IWC, and De reveal again that one of the three parameters can be derived from the other two from the previously built lookup tables at different temperature ranges. The two lookup tables for the Ze-IWC relationships with different De at two temperature ranges −50°C < T < −40°C and −40°C < T < −25°C) shown in Figure 8 are used to estimate ice cloud De. The estimated ice cloud De agree well with the De calculated using the ice particle size distributions (Figure 9). The relative errors for the two temperatures ranges of −50°C < T < −40°C and −40°C < T < −25°C are less than 32% and 24%, respectively. The RMS of estimated De are about 8 μm and the correlation coefficients between the estimated ice cloud De and the De calculated using the ice particle size distributions are over 94%.
4. Summary and Conclusions
 The effect of ice particle habits on Ze-IWC relationships is investigated using six different ice habits including hexagonal solid and hollow columns, plates, 3D bullet rosettes, aggregates, and droxtals. The Ze-IWC relationships for ice clouds composed of these habits are derived by the calculated Ze and IWC from 1119 measured particle size distributions obtained from a variety of field campaigns. The Ze-IWC relationships obtained for these individual habits show distinct differences. For a given Ze, the IWC vary in a factor of 1.5–2.0 for ice cloud composed of different habits, and in particular, the variations in IWC are larger when the Ze are negative than when the Ze are positive.
 The Ze-IWC relationship has been found to be sensitive to the variability in the ice particle spectrum [Atlas et al., 1995; Schneider and Stephens, 1995; Brown et al., 1995; Aydin and Tang, 1997; Liu and Illingworth, 2000; Sassen et al., 2002]. In the present study, on the basis of the 1119 measured measurement particle size distributions, the effect of the particle effective size (De) on the Ze-IWC relationships is investigated by deriving the Ze-IWC relationships for different ranges of De. The IWC generally increases with decreasing De for a given Ze. The dependence of Ze-IWC relationships on De shows a regular feature, which may be potentially useful for estimating De from observed Ze.
 The effect of temperature on the Ze-IWC relationships reveals that the inclusion of temperature in Ze-IWC relationship has no significant improvement for estimating IWC. This is also revealed by the relationships between temperatures and De derived from 1119 data sets measured for ice clouds. However, for a given temperature range, the dependence of the Ze-IWC relationship on De is pronounced. This provides an opportunity to obtain De from the Ze-IWC relationship. The Ze-IWC relationship is derived for different De for the two temperature ranges. The dependence of the Ze-IWC relationship on the effective particle size within a given temperature range is pronounced, and may be useful for inferring the cloud effective particle size from a Ze-IWC relationship. It is difficult to apply the Ze-IWC relationships derived for different De within given temperature ranges to operational radar retrieval because the information about De (for estimating IWC) or IWC (for estimating De) is needed. However, the information can be provided by observations made by other active and passive sensors. Moreover, these relationships can be used to simulate radar Ze of ice clouds simulated from the weather forecasting, mesoscale, climate models that output IWC, De, and temperatures.
 The authors thank B. T. Draine and P. J. Flatau for providing their well-documented DDA model. The authors also thank the three anonymous reviewers for constructive comments and suggestion. Ping Yang's research is supported by a National Science Foundation (NSF) grant (ATM-0239605).