## Introduction

The ice water path (IWP), defined as the integral of the ice water content (IWC) through the depth of an ice cloud layer, is a quantity of considerable importance in climate studies because it is critical for determining cloud absorption, optical depth (Platt and Harshvardhan 1988), albedo, and emissivity (Stephens 1980). In this study, we investigate the relationship between cloud optical depth in visible wavelengths (*τ*_{υ}) and the IWP based on in situ measurements. This relationship is important in climate model studies and for deriving IWP from radar and satellite retrievals of cloud optical thickness.

Methods developed for determining IWP fall into three primary categories. Measurements from remote sensors collecting cirrus cloud data at two or more wavelengths have been converted into IWP based on assumed or measured vertically and horizontally homogeneous ice particle size distribution models for different cloud types and geographical locations. For example, for cirrus clouds that are not optically thick, Liou et al. (1990) showed that radiances measured at two infrared wavelengths (6.5 and 10.5 *μ*m) could be used to retrieve IWP. Wu (1987) also showed that measurements from one near-infrared channel and a microwave radiometer at a higher frequency could be used for IWP estimation. For optically thin clouds, visible sensors can provide an estimate of *τ*_{υ}, which may then be converted to IWP, given an estimate of effective particle size (Ebert and Curry 1992; Lin and Rossow 1996; Wang and Sassen 2002). For thick precipitating ice clouds, brightness temperatures at 37 and 85 GHz were found useful in estimating the IWP (Vivekanandan et al. 1991; Weng and Grody 1994). Matrosov et al. (1992) and Mace et al. (1998) have developed techniques that use a combination of radiometer measurements at infrared (10–20 *μ*m) wavelengths and radar reflectivity measurements to derive cloud-layer-averaged values of IWC, IWP, and estimates of *τ*_{υ} for optically transparent cirrus clouds.

A second approach for obtaining IWP has been to consider a cloud as a layer of saturated air ascending over a range of velocities and temperatures, with the IWP given by the difference between the rates of buildup of ice condensate and ice particle sedimentation (Heymsfield and Donner 1990). No information on the properties of the particle size distributions (PSDs)—or optical depth—is required or obtained. A third approach utilizes empirical relationships between IWC or IWP and a more easily observed quantity, such as radiative attenuation (Novosel'tsev 1962), temperature (Heymsfield and Platt 1984; Kosarev and Mazin 1989), or radar reflectivities (Heymsfield 1977; Sassen 1987).

Recent advances in satellite radiance measurements made possible by the Moderate Resolution Imaging Spectroradiometer (MODIS) instrument on the National Aeronautics and Space Administration (NASA) *Terra* and *Aqua* satellite platforms have provided opportunities to retrieve cloud microphysical properties, including IWP, with higher spatial resolution and better accuracy than in the past. (Being passive sensors, satellite radiometers can still give only one homogeneous quantity). MODIS has 36 discrete spectral bands and greatly improves upon earlier National Oceanic and Atmospheric Administration (NOAA) satellite radiometers—for example, the Advanced Very High Resolution Radiometer (AVHRR)—both in spatial and spectral resolution. Through the use of retrieval algorithms that consider ice and liquid water clouds as vertically inhomogeneous layers with realistic ice particle size distributions and habits (Yang et al. 2001; Nasiri et al. 2002; Platnick 2000), MODIS measurements of solar reflectance in visible and near-infrared atmospheric window bands can be used to retrieve the visible ice cloud optical thickness and the effective radius (*r*_{e}) or effective diameter (*D*_{e} = 2*r*_{e}) more accurately than in the past. The effective radius is essentially proportional to the IWC divided by the extinction coefficient, and is defined in different ways by different researchers. Given that two MODIS instruments are currently operational, new methods developed to exploit the instrument's capabilities could lead to improvements in IWP retrieval and the use of these data in general circulation models (GCMs).

This study is motivated by the need to produce reliable estimates of IWP from satellite measurements from a combination of visible and near-infrared wavelengths or from ground-based radars. These satellite or ground-based products are critical for use in GCM studies, and to provide constraints on the estimates of IWP retrieved from planned spaceborne cloud radar measurements. Our approach uses in situ observations of particle size distributions and habit information through the depths of midlatitude and tropical ice clouds to develop an understanding of the factors that influence the relationships between IWP, *τ*_{υ}, and *r*_{e}. We begin by describing our dataset in section 2. In section 3, we compare the observed values of *τ*_{υ} and IWP to those retrieved using the earlier methods. In section 4, we detail the factors that affect the relationship between *τ*_{υ} and IWP and develop a means of deriving *r*_{e} from remote sensing observations. A summary and some conclusions are presented in section 5. Appendix A provides a list of the symbols used in the paper and their descriptions.

## Data collection

Ice particle size distributions and low- and high-resolution ice particle images were collected during Lagrangian spiral descents and balloonborne ascents through ice cloud layers that formed in association with synoptic-scale lifting (midlatitude) and deep convection (Tropics). During the Lagrangian descents, the aircraft descends at about the mean fall velocity of the particles in an attempt to characterize particle evolution in the vertical, as discussed in Field and Heymsfield (2003) and the references cited therein. This type of flight pattern allows us to examine the underlying factors that control the relationship between *τ*_{υ} and IWP.

Aircraft measurements were acquired in midlatitude cirrus by the National Center for Atmospheric Research (NCAR) King Air during the First International Satellite Cloud Climatology Project (ISCCP) Regional Experiment (FIRE I) near Madison, Wisconsin, in 1986. Data were also acquired in midlatitude cirrus by the University of North Dakota Citation during an Atmospheric Radiation Measurement (ARM) intensive observation period during the spring of 2000 near Lamont, Oklahoma, and in tropical ice cloud layers during the Tropical Rainfall Measuring Mission (TRMM) field program near Kwajalein, Marshall Islands, during August and September 1999. Balloonborne measurements were acquired in midlatitude cirrus using NCAR ice-crystal replicators (Miloshevich and Heymsfield 1997) during the Second FIRE (II) program near Coffeyville, Kansas, in November and December 1991.

The conditions encountered during the approximately 30-min-sampling intervals for the aircraft descents and approximately 15-min intervals for the replicator ascents through cloud layers, are summarized in Table 1. The duration of each in situ sample interval is indicated in the table. Temperatures for the midlatitude ice clouds ranged between −20° and −65°C. The airborne measurements extended to −53°C, and the replicator measurements to −65°C. The six tropical cloud observations occurred at temperatures between 0° and −50°C. Using several probes, including a forward scattering spectrometer probe (FSSP), King probe, and Rosemount icing probe, we did not detect liquid water at subfreezing temperatures in the tropical clouds above the probes' detection limit of about 0.01–0.05 g m^{−3}. The average cloud thickness sampled for all clouds combined was 3.0 ± 1.0 km. Several of the tropical clouds extended into rain regions above 0°C, and although they were sampled, those portions are not considered here. This omission will have no impact on this study, because we are examining the relationship between optical depth and ice water path and are not evaluating cloud radiances that would be affected by the presence of liquid water. The diameters of the loops of the individual Lagrangian spirals were approximately 5 km. Horizontal inhomogeneity was observed across the loops of individual spirals, as described in detail by Heymsfield et al. (2002a) and elaborated upon in section 3a. Although there are issues related to temporal evolution during the time periods required for the Lagrangian spiral descents, these are secondary for the clouds studied here (as discussed in Field and Heymsfield 2002).

On the aircraft, size-spectra measurements were obtained with imaging probes. For sizes from about 50 to 1000 *μ*m, Particle Measuring Systems, Inc., (PMS) 2D-C probes were used. The 2D-C resolution was 25 *μ*m for FIRE I and 33 *μ*m for the other field campaigns. For sizes from 1000 to more than 3000 *μ*m for FIRE I and the ARM campaigns, PMS 2D-P probes were used. The 2D-P probe resolution was 100 *μ*m for FIRE I and 200 *μ*m for the ARM campaign. The PSDs in sizes from 1 to more than 30 mm for the TRMM field campaign were obtained from a Stratton Park Engineering Company (SPEC) high-volume precipitation spectrometer (HVPS) probe; probe resolution was 0.2 mm. A SPEC cloud particle imager (CPI) probe provided imagery with 2-*μ*m resolution over the range of sizes from about 20 to 2000 *μ*m for the TRMM and ARM campaigns. Average size distributions were measured over 5–7-s intervals, or about 1 km of horizontal flight distance. Further discussion of the sample volume of each of these probes, the data processing techniques, and how the size distributions based on the different probes were eventually merged is provided in Heymsfield et al. (2002a).

The replicators used in FIRE II yielded size distributions and particle habit imagery with a resolution of about 2 *μ*m from about 10 to between 500 and 1000 *μ*m in size. The replicators could measure size distributions and yield particle images quite reliably down to the smallest range, 10–20 *μ*m in diameter. The aircraft forward scattering spectrometer probes (sizing from 2 to 30 *μ*m) could not measure small particle concentrations reliably because of possible breakup in the probe inlet and are not used. Therefore, the replicators were used to investigate the contribution that small particles make to bulk cloud microphysical properties, including IWC. The CPI probe produces detailed particle imagery for these sizes, but its accuracy in measuring size distributions is not yet known; therefore, we did not use the CPI to evaluate the contributions of small particles. The continuous replicator observations were averaged over vertical distances of about 300 m to obtain statistically significant particle size distributions, as well as the area of each of the sampled particles, which is approximately their cross-sectional area (Heymsfield and Miloshevich 2003).

Two key parameters needed for our analysis are the total cross-sectional areas of the particle populations, and the IWC. Cross-sectional areas of particles in the horizontal—their fall mode and approximately how they would be viewed from a satellite—were measured directly by the various probes, above the probes' size detection thresholds. A good match in the distribution of area per unit volume with particle size (*D*) was found between the high-resolution CPI and low-resolution 2D-C and 2D-P probes (Heymsfield and Miloshevich 2003), justifying the use of the 2D probe data to obtain particle area and the total cross-sectional area of the particle population per unit volume (*A*_{c}) above their detection threshold. Even so, the values of *A*_{c} from these probes are underestimated, because of the probes' minimum detection size threshold. Observations of small particles by the replicator were especially relevant for estimating the contributions of small particles to the total values of *A*_{c}, but a number of other means are used throughout this paper to estimate the contributions by small particles.

*σ*) can easily be found from twice the

*A*

_{c},where

*A*is the area of particles of size

*D.*The ice water contents given by the distribution of ice mass with size integrated over

*D,*fromwhere

*N*is the concentration,

*m*is the mass,

*D*

_{min}is the probe size detection threshold, and

*D*

_{max}is the size of the largest particle in the particle size distribution. We discretized the size distributions into a series of bins and numerically integrated over all bins rather than use a parameterized form of the size distribution. The bin centers correspond approximately to the resolutions of the probes. The mass extinction coefficient

*K,*a measure of the relative importance of the extinction coefficient as compared with the IWC, is

Ice particle mass is found in the following way. In general, ice particle mass is given by *m* = (*π*/6)*ρ*_{e}*D*^{3}, where *ρ*_{e} is the ice particle effective density. For many years, mass has been represented as a power law of the form *m* = *βD*^{χ}, where *β* and *χ* are habit-dependent coefficients. The power-law method circumvents the need to obtain *ρ*_{e}. Heymsfield et al. (2002b) sought to improve the estimates of *β* and *χ* by employing the area ratio, *A*_{r}, where *A*_{r} = *A*/[(*π*/4)*D*^{2};cb. Here, *A*_{r} provides intrinsic information on particle shape and can be used to accurately estimate the effective ice particle density (Heymsfield et al. 2002b) if the particle habit falls into one of the more common ice particle shapes, including aggregates. In their approach, *ρ*_{e} is represented in terms of *A*_{r}, yielding *m* = (*π*/6)*ρ*_{e}*D*^{3} = (*π*/6)*k**A*^{n}_{r}*D*^{−α}*D*^{3}, where the terms *k,* *n,* and *α* are habit-dependent coefficients derived from modeling calculations and ice particle collections at the ground. These coefficients have been validated by comparison to direct measurements of IWC, radar reflectivity, and fall velocities in Heymsfield et al. (2002a, 2002b). For the midlatitude clouds, the absolute accuracy of the estimate of IWC is better than ±20%, as ascertained from direct measurements of IWC (Heymsfield et al. 2002b), and for the tropical clouds it is better than ±30% (Heymsfield et al. 2002a), as ascertained through a less direct comparison with radar reflectivity observations. As in Heymsfield (2003, hereinafter HO3) for both midlatitude and tropical observations, the nominal values of these coefficients are *k* = 0.07, *n* = 1.5, and *α* = −0.5, all in cgs units. These coefficients are adjusted above 0.1 cm to account for a decrease in *ρ*_{e} with size resulting from ice particle aggregation.

Average coefficients *β* and *χ* may be determined from our approach for reference to the coefficients found in earlier studies. Heymsfield et al. (2002a) have formulated *A*_{r} in terms of *D,* where *A*_{r} = *aD*^{b}; *a* and *b* are not constants but are related to the slope of the size distribution. Therefore, *m* = (*π*/6)*ka*^{n}*D*^{(3+α+bn)} = *βD*^{χ}. Average values of *β* and *χ* can be determined by examining all measured size distributions obtained from the various field experiments. The results for the midlatitude dataset differed from those obtained from the tropical dataset, with average values of *m* = 0.0065(±0.003)*D*^{2.25±0.015} for the midlatitude dataset, and *m* = 0.0038(±0.002)*D*^{1.96±0.014} for the tropical dataset. Preliminary investigation indicates that the difference in the coefficients is due to particle habit (area ratio) differences. For reference to earlier studies, planar polycrystals in Mitchell (1996) have *β* = 0.007;th39 and *χ* = 2.45 (cgs), whereas the habit-independent relationship in Brown and Francis (1995) yields *β* = 0.002;th94 and *χ* = 1.9. The former relationship returns lower values of ice particle mass, whereas the latter relationship returns comparable values of mass at sizes below 1000 *μ*m and somewhat lower values above this size.

*h*) at cloud base (

*h*

_{base}) to top (

*h*

_{top}):The size distribution and area data obtained from each 1 km of flight distance during each Lagrangian spiral descent are summed for the entire spiral to obtain

*τ*

_{υ}and IWP values, and are found by summing the replicator observations for each ascent from each 300-m (average) size distribution.

## Observations

In this section, we present calculations of *σ,* IWC, *τ*_{υ}, and IWP for our sample size of 13 midlatitude and 6 tropical cases. There are a total of 889 midlatitude and 1125 tropical size distributions used in this study. Although this sample size is small, it is far larger than the number of samples used in earlier studies, and has the added benefit that it contains data from the top to the bottom of cloud layers. Furthermore, the ranges of the IWC and *σ* values considered here are larger than previous studies. The IWCs span four orders of magnitude, from 10^{−4} to 1 g m^{−3}, and the *σ* values also range over four orders of magnitude, from 10^{−5} to 0.1 m^{−1}. The results of this analysis are compared to those from earlier observations, as a means of assessing the generality of these results and whether our results are insightful. For reference in the following discussion, the observed particle habits are primarily bullet rosettes and aggregates of bullet rosettes for the midlatitude observations. The habits are primarily complex single crystals and aggregates, some of which are rimed, for the tropical cases. Examples of images of the observed particles appear in Heymsfield et al. (2002a,b).

### Mass extinction coefficient

From Eqs. (4) and (5), the integration of *σ* and IWC over height yields values for *τ*_{υ} and IWP. If the the mass extinction coefficient *K* [Eq. (3)] is constant with height, then *τ*_{υ} and IWP are directly proportional. If *K* is not constant with height, a more complex relationship between *τ*_{υ} and IWP will be found. This section examines the dependence of *K* on height, and on other variables that depend on height.

Figures 1a and 1b show *K* over the course of measurements through two midlatitude cloud layers, one of which was a cold case sampled by a replicator, and two tropical ice cloud layers. These cases bound the range of temperatures from our observations. One of the two tropical cases, the case on 22 August 1999 is different than the other tropical cases in that the data were collected in an anvil in close proximity to convection, resulting in the lowest values of *K* found in the tropical dataset. Each data point from the aircraft observations represents a vertical depth of about 30 m and each replicator data point represents about 300 m in the vertical. For the aircraft spirals, there were between 8 and 14 loops over the course of the descents from cloud top to bottom, with each loop comprising about 300 m of vertical descent. The values of *K* did not vary much during these small changes in height associated with the loops of the spirals (Figs. 1a and 1b), indicating that variability in *K* during the individual loops of the spirals is relatively small. The decrease in magnitude of *K* from the top to the bottom of a spiral is, therefore, usually greater than over the course of the individual loops of the spirals. Horizontal inhomogeneities, therefore, will not interfere with efforts to develop a parameterization of the trends in *K* with height or temperature.

The values of *K* are found to decrease downward, indicating that small particles that influence *σ* more than IWC are more prevalent toward the upper parts of the clouds, whereas large particles are more prevalent toward the cloud base. This effect was most pronounced in the replicator case, which had the largest range in values of *K.* What is also surprising, from Figs. 1a and 1b, is that the values of *K* span a very narrow range from about 0.02 to 0.06 m^{2} g^{−1}.

Figures 1c and 1d show how the values of *K* change with height above cloud base (Δ*h*) for one aircraft spiral from Figs. 1a and 1b. The trend noted in each figure for *K* to increase with (Δ*h*) is typical of all cases. The values of *K* increase approximately linearly with (Δ*h*), with *K* increasing by an order of 0.003 m^{2} g^{−1} km^{−1}.

Also shown in Figs. 1c and 1d are linear curve fits between *K* and Δ*h* that fit the data quite well. The form of these curve fits are such that they can be readily integrated over height *h* to produce a relationship between *τ*_{υ} and IWP, with two terms: a constant factor and a factor that is proportional to Δ*h.* This form is much like earlier relationships developed between *τ*_{υ} and IWP (e.g., Ebert and Curry 1992). The details of the integration process will be discussed in section 3c.

As shown in Figs. 2a and 2b, for the same cases as Figs. 1c and 1d, *K* also decreases approximately linearly with increasing temperature over the course of the spirals. Fitted expressions indicated in each panel also justify the use of two terms to represent the relationship between *τ*_{υ} and IWP, and can also be readily integrated.

The example shown in Fig. 2b represents our coldest tropical case but is anomalous in that particle sizes reached almost 10 mm at a temperature of −50°C, with correspondingly low values of *K* observed throughout the cloud layer. For this reason, our expectation is that our values for *K* from the tropical observations for temperatures below −40°C (which are only from this case) apply to optically thick clouds in the vicinity of convection.

The variation of *K* with temperature for a larger portion of the dataset is shown in Fig. 2c. The trends with temperature mirror those shown in Figs. 2a and 2b. To reduce clutter in this figure, only each fourth point from the total of 2000+ particle size distributions is plotted, and curves fitted to the data are listed and plotted. The values of *K* for the midlatitude points tend to be higher in overlapping temperature ranges, and the increase in *K* with decreasing temperature appears to be larger for the midlatitude than tropical points. This feature is apparent in the curve fits plotted and listed in the figure: *K* ∝ −3.1 × 10^{−4}*T* for the tropical points versus *K* ∝ −7.4 × 10^{−4}*T* for the midlatitude points. Note that the curve for the tropical data in Fig. 2c is extrapolated to make it more visible. Note also that the values for *K* for the case on 22 August 1999 are anomalously low, apparently due to the proximity to convection. For this reason, the curve fit for the tropical cases in Fig. 2c does not include this case.

It is important to point out that the tropical observations apply to optically thick clouds with high IWCs. The relationship between *K* and IWC is discussed in detail in section 3c.

Our finding that *K* increases with increasing height and decreasing temperature can be compared to findings from earlier cirrus studies. Most earlier studies have either examined IWC or *τ*_{υ} independently, or have calculated the variation of the effective radius (*r*_{e}) with height under the assumption that *K* is proportional to 1/*r*_{e} plus a constant. A tendency for *K* to increase with height was observed in two midlatitude cirrus layers by Francis et al. (1994), with values for *K* similar to those found here. Conversely, for three midlatitude cloud layers sampled by Francis (1995) and one midlatitude layer sampled by Kinne et al. (1992), this tendency was not found, although the values of *K* in these studies were comparable to ours. In these two studies, large horizontal variability in the cloud microphysical properties was noted. For tropical clouds, Heymsfield and McFarquhar (1996) found a general tendency for *K* to increase with decreasing temperature, although there was considerable variability. Because the studies cited in this paragraph are based on horizontal legs rather than Lagrangian spiral descents, we conclude that it is difficult to assess the vertical profile of *K* based on an examination of average values from horizontal legs.

The relatively rapid increase in *K* at the lowest temperatures in Fig. 2c suggests that small particles have an increasingly large effect on *K* with decreasing temperatures. The role of small particles on the vertical distribution of *K* can be assessed by using the calculated vertical distributions of IWC, *σ,* and *K* from the three replicator ascents. Figure 3a shows the ratio of the IWC in small particles, subjectively defined here as those smaller than 50 *μ*m, to the total IWC. For each replicator case, a significant percentage of the total IWC, between 20% and 80%, is contained in the upper 300 m of the cloud layer. The secondary peak noted in the ratio for the case on 26 November 1991 at a temperature near −40°C is located at the top of a second cloud layer (as seen with a ground-based lidar), where numerous small ice crystals were observed. The percentages of the IWC present in small particle sizes rapidly diminishes as the cloud base is approached, accounting for less than 10% of the total IWCs. As an average for the cloud layer, small particles contribute between 2% and 12% of the total IWC (see legend in Fig. 3a), with increasing amounts associated with decreasing temperatures.

Francis et al. (1994) also concluded that a large proportion of the IWC was contained in small crystals near the tops of two midlatitude cirrus clouds that they sampled at temperatures near −40°C; this proportion rapidly diminished below cloud top. A similar result is noted in the cirrus measurements of Arnott et al. (1994), where a large proportion of the IWC in small sizes, as derived from replicator measurements, was contained near the very top of the layer, with a rapid reduction in the proportion below cloud top.

The contribution by small particles to the vertical distribution of *σ* (Fig. 3b) is similar to that observed for IWC, with small particles in the secondary layer on 26 November 1991 producing significant contributions, and the largest contribution found for the coldest case. As expected in relative terms, small particles contribute more to *σ* than to IWC. This result is similar to the observations in a cirrus layer by Arnott et al. (1994), who showed that the total particle area was dominated by small particles near cloud top, with a markedly decreasing contribution by small particles below cloud top.

The effect of small particles on *K* is examined by comparing *K* in particle sizes above 50 *μ*m to those observed over the full range of replicator sizes (Fig. 3c). It is noted that small particles lead to a 20% increase in the value of *K* near cloud top. However, because small particles contribute both to IWC and *σ,* omission of small particles only leads to a small reduction in the layer-averaged value of *K.*^{1}

In appendix B, the McFarquhar and Heymsfield (1997) CEPEX size-spectra parameterization is used to further estimate the effects that particles smaller than the detection threshold of the 2D-C probe might have had on *σ,* IWC, and *K.* Given a value for the IWC measured by the 2D-C probe and temperature (for *T* < −20°C), the total IWC, value of *σ* in 2D-C sizes, and total *σ* can be estimated. This analysis suggests that the values of *K* for the midlatitude cases (aside from the replicator cases) with low values of *τ*_{υ} may be underestimated by 20%–50%, but the foregoing discussion from the replicator data (e.g., Fig. 3) suggests that the underestimation is much smaller.

We will now return to an examination of the factors that influence the vertical distribution of *K.* The median mass diameter, *D*_{m}, the size that divides the distribution of mass with size in half, is a useful parameter that describes the mean mass–weighted size of a particle size distribution. The value of *D*_{m} for each particle size distribution was found numerically, although *D*_{m} can also be found analytically (Mitchell 1991). As noted in Figs. 4a and 4b, there is a well-defined inverse relationship between the mass extinction coefficient and *D*_{m}. The decrease in *K* with decreasing height within ice cloud layers (Figs.1a–d) indicate that particle mass shifts to larger sizes within the cloud layer. The fitted linear inverse relationships between *K* and *D*_{m}, plotted and listed in the figures, indicate that *τ*_{υ} and IWP can be related through *D*_{m}.

For comparison with earlier studies, we can assess the range of *r*_{e} values observed in our study, again, using the parameterized forms of the PSDs and particle density. The quantity *r*_{e} has formed the basis of *τ*_{υ} versus IWP relationships developed in earlier studies, and there are many definitions used in the literature. The habit-independent definitions of *r*_{e} or *D*_{e} given by Fu (1996), *r*_{ge} or *D*_{ge}, respectively, were developed to preserve *A*_{c} and IWC. These definitions are used here because the observed particle habits comprised a variety of complex shapes. According to this definition, *r*_{ge} = [*ρ*_{i})](IWC/*A*_{c}) = [*ρ*_{i}0.5*K*)], where *ρ*_{i} is the density of solid ice. There should be an inverse relationship found between *r*_{ge} and *K,* and this inverse dependence is noted in Figs. 4c and 4d. The *r*_{ge} values in the figures range between 20 and 100 *μ*m.

Also plotted in Fig. 4d are *r*_{e} values from a video ice particle sampler (VIPS) that was used during CEPEX to measure particle size distributions between approximately 10 and 90 *μ*m in maximum dimension, combined with data from the 2D-C probe above 90 *μ*m. The resulting *r*_{e} values are smaller than are observed here, a result that is not surprising because the particles were smaller and the IWCs were lower, as shown in section 3C.

HO3 showed that a single gamma-type size distribution fit the size distributions from the imaging probes reasonably well, although these do not include particles smaller than about 30 *μ*m. The gamma distribution takes the form *N* = *N*_{0}*D*^{μ}*e*^{−λD}, where *N* is the concentration, *N*_{0} is the intercept, *μ* is the dispersion, and *λ* is the spectral slope of the distribution. The fit coefficients are derived by a moment-matching method that minimizes the errors in the first, second, and sixth moments of the distributions. Decreasing values of *λ* are generally found to be associated with increasing temperatures. This tendency is also shown by Platt (1997) and others, who used exponential size distributions (*μ* = 0) to represent cirrus and frontal cloud size distributions. The term *μ* is related to *λ,* with negative values of *μ* to be associated with low values of *λ* and positive values of about 2 or 3, associated with large values of *λ.*

Figures 5a and 5b show *K* plotted as a function of *λ.* Curves fitted through the 2000+ size distributions are plotted in the figures, with the fitted equations shown. There are well-defined relationships between *K* and *λ.* At large values of *λ,* where the size distributions are generally narrow with a relative abundance of small particles and extinction dominates over IWC, the values of *K* are an order of 0.06 *m*^{2}/*g.* At small values of *λ,* characterized by relatively broad size distributions and relatively numerous large particles, the *K* values tend toward 0.02 *m*^{2}/*g.* Note also that the size distributions from the tropical observations are broader (smaller values of *λ*) than those from the midlatitude observations; this is not surprising given the temperature ranges considered in these datasets. The relationship between *K* and *λ* are well represented by both power-law and linear equations plotted and listed in the figures. The relationship between *K* and *λ* is discussed in detail in section 4.

### Extinction–ice water content

This section examines the relationship between *σ* and IWC, as a means of further assessing the factors that contribute to the relationship between *τ*_{υ} and IWP. Because earlier studies have attempted to directly relate *σ* and IWC, it is desirable to compare our results to these earlier studies.

Our observations of the relationship between *σ* and IWC for the midlatitude and tropical spirals are plotted in Figs. 6a and 6b. The *σ* and IWC values cover four orders of magnitude. Power-law curves fitted to the observations, both plotted and listed in the figure, indicate that *σ* ∝ (IWC)^{0.9}.

Because the replicator collected information on small particles, a separate curve is plotted in Fig. 6a to represent the *σ* versus IWC data from that probe. As noted in the figure, there is very little difference between the curve from the replicator and the curve from the full dataset.

Platt (1997) parameterized *σ* in terms of IWC for midlatitude frontal and cirrus clouds, using particle shapes modeled after cylinders to represent columns, single bullets, and size distributions. Platt's curve is plotted on the curve for the midlatitude observations in Fig. 6a, where it is noted that the slope of his curve (0.68) is lower than the slope of ours (about 0.9). The Platt parameterization fits our observations quite well over a fairly wide range of IWCs, although the higher values of *σ* and low IWCs indicate that there were more smaller particles represented in Platt's relationship than are found in our replicator observations.

The lightly shaded points in Fig. 6b are from the case on 22 August 1999 that was sampled in an anvil adjacent to convection and it is clearly different from other tropical cases because it has lower values of *σ* for a given IWC. The lower relative values of *σ* was a result of the presence of very large particles throughout, even near cloud top at −50°C. This case is highlighted in Fig. 1b (light shaded points), because the values of *K* fell below those of the other case shown.

Also plotted in Fig. 6b is a curve from Heymsfield and McFarquhar (1996), who used 2D imaging-probe data to derive a relationship between *σ* and IWC for the tropical ice clouds they observed during CEPEX. This curve overlies the curve fitted to the tropical dataset, and although not shown, overlies the midlatitude dataset as well.

In Fig. 6b, very few of our data points fall below an IWC value of 0.01 g m^{−3}, and little guidance is available from our dataset to characterize the *σ* versus IWC relationship for tropical clouds with low IWCs. For this reason, we have included data from the VIPS that was used during CEPEX to measure particle size distributions between approximately 10 and 100 *μ*m in maximum dimension. The 53 data points from the combination of VIPS and 2D-C data that covered the size range from 10 to above 1000 *μ*m are plotted in Fig. 6b, and a line is fitted to the data. Although there might be a tendency for the *σ* versus IWC curve to have a slightly steeper curve than the tropical points observed here, this is not borne out by the data. What is noted is more variability. A further examination of the relationship between *σ* and IWC based on the McFarquhar and Heymsfield (1997) parameterization is presented in appendix B.

In most earlier studies of the relationship between *τ*_{υ} and IWP, it was assumed that the particles had the same shape as hexagonal columns. Using the relationship between length and width with size *D,* the ice densities assumed for hexagonal columns in these earlier studies (Ebert and Curry 1992; Fu 1996), and our measured size distributions, we have rederived the relationship between *β* and IWC, assuming that all particles were hexagonal columns. To avoid an additional figure, we have fit curves to the resulting midlatitude and tropical observations, with the results plotted in Figs. 6a and 6b (“hexagonal column model”). The curves are clearly shifted to the right of the curves fitted to the results for the actual particle habits. This result is readily explained. The extinction cross sections derived for hexagonal columns have similar values to those found for the particles observed in our study for the same particle size. However, hexagonal columns have about twice the mass of the bullet rosettes and aggregates we observed for the same particle size. The result is that for the same particle size, the IWC for a given extinction coefficient for hexagonal columns is about twice the value found in our data.

Fig. 6c shows the mass extinction coefficient as a function of IWC for all of our datasets combined. It is noted that there is a well-defined inverse relationship between *K* and IWC that fits the CEPEX-combined VIPS and 2D-C data as well. This figure clearly shows that temperature alone as in Fig. 2c is not the only factor that influences *K.*

### Optical depth–ice water path

Using the measured particle size distribution and area data, *τ*_{υ} and IWP were calculated for the midlatitude and tropical datasets. Our vertical profiles allow us to calculate *τ*_{υ} and IWP directly and investigate how they are related. This has not been possible in most earlier studies using in situ data because horizontal rather than vertical sampling profiles were emphasized. Although our estimates of *τ*_{υ} and IWP are not true values, because horizontal variability over the course of the spirals required that spiral average values be obtained, these data will allow us to reach our goal of deriving general relationships.

The *τ*_{υ} and IWP values shown in Fig. 7a cover wide ranges, from 0.4 to 40 g m^{−2}, and from 10 to 1000 g m^{−2}, respectively. The values of *τ*_{υ} for the tropical data are all very large, between 20 and 30, signifying very optically thick clouds. A power-law least squares curve fitted to the midlatitude *τ*_{υ} vs. IWP data shown in the figure has a power of about 0.8. The finding that the power in the relationship between *τ*_{υ} and IWP is less than unity signifies that *σ*/IWC is not constant with height, temperature, or IWC, mirroring the findings from Figs. 1, 2, and 6. Because there are no values of *τ*_{υ} for the tropical dataset that are below about 20 and the range of the tropical *τ*_{υ} values is small, no curve is fitted to that data. A composite curve produced by combining the midlatitude and tropical datasets is shown in the figure (“All”), but this relationship may not account adequately for low optical depth tropical anvil clouds that may be dominated by aggregates and rimed crystals. Nevertheless, the All curve may be useful in some instances because it provides a relationship between *τ*_{υ} and IWP without the need for other variables.

Also shown in Fig. 7a is a curve fit to all of the data except for those cases where the values of *τ*_{υ} fall below 1.0. The argument is that for these cases, owing to their relatively low IWCs and the absence of small particle data, *τ*_{υ} may be too low. This curve differs little from the curve fitted to all of the data.

*τ*

_{υ}and IWP in terms of the effective radius, as has been done in earlier studies. Linear curves have been fitted between the ratio

*τ*

_{υ}/IWP and the inverse of the layer mean effective radius (

*r*

_{ge}

*r*

_{ge}

*g*

_{0}and

*g*

_{1}/

*g*

_{0}for the midlatitude observations (see also Fig. 4c) are 0.012;th56 and 0.725/

*g*

_{0}= 57.7 (correlation coefficient of curve fit is 0.75), respectively, and for the tropical clouds (see Fig. 4d) are 9.31 × 10

^{−3}and 0.67/

*g*

_{0}= 71.7 (correlation coefficient of 0.75), respectively.

*r*

_{ge}and IWP from our data. This allows Eq. (6) to be represented solely in terms of IWP. The relationships are found to have the forms

*r*

_{ge}

^{0.29}(midlatitude), with a correlation coefficient of 0.68, and

*r*

_{ge}

^{0.56}(tropical), with a correlation coefficient of 0.75 These equations are then inserted into Eq. (6). The resulting curves in Fig. 7a, one for the midlatitude observations (solid gray line) and one for the tropical observations (solid black line), closely fit the observations. The relationship fit to the combined midlatitude and tropical observations covers a wide range of IWP values and fit the observations as well:with a correlation coefficient of 0.81.

*r*

_{e}

*r*

_{e}, we have fitted a relationship between

*r*

_{e}and IWP separately for the midlatitude and tropical datasets. These equations are then input into Eq. (8), from which

*τ*

_{υ}and IWP are related directly. The resulting curves, plotted in Fig. 7b, fit the observations reasonably well.

Wang and Sassen (2002) used a relationship among *τ*_{υ} and IWP of a similar form, *τ*_{υ} = IWP(−2.936 × 10^{−4} + 1.2725/*r*_{ge}*σ,* IWC, and *r*_{ge} developed by Fu (1996). The Wang and Sassen equation may also be compared with our *τ*_{υ} versus IWP data shown in Fig. 7b, using the *r*_{ge} and IWP relationships developed for our datasets. The curve, based on their equations, falls on top of our tropical curve and close to our midlatitude curve.

*τ*

_{υ}and IWP to include dependent variables other than

*r*

_{e}. The units of IWP in the following discussion are in grams per meter squared. Figures 1c and 1d show how the values of

*K*change with height above cloud base (Δ

*h*=

*h*−

*h*

_{base}) for two aircraft cases. In these figures, equations of the form

*K*≈

*c*

_{0}+

*c*

_{1}, where

*c*

_{0}and

*c*

_{1}are constants, are given. With the cloud-base height (

*h*

_{base}) as a reference point,

*K*=

*σ*/IWC can be integrated over height throughwhere, as noted in Figs. 1a and 1b, the coefficients

*c*

_{0}and

*c*

_{1}are case dependent. In Eq. (9), the average height above cloud base in kilometers, Δ

*h*/2, can be extracted from the integral with little loss of accuracy, because

*σ*/IWC varies approximately linearly with height. On average, for the midlatitude clouds,

*c*

_{0}= 0.030 ± 0.008 and

*c*

_{1}/

*c*

_{0}= 0.0752 ± 0.118; for the tropical clouds,

*c*

_{0}= 0.0183 ± 0.002 and

*c*

_{1}/

*c*

_{0}= 0.0815 ± 0.035. The relatively small standard deviations suggest that Eq. (9) may provide a useful indicator of the

*τ*

_{υ}–IWP relationship when cloud thickness is known.

*τ*

_{υ}and IWP can be related through the mean cloud temperature,where

*T*is in degrees. On average for the midlatitude clouds,

*d*

_{0}= 0.025 ± 0.016 and

*d*

_{1}/

*d*

_{0}= −0.034 ± 0.056, and for the tropical clouds,

*d*

_{0}= 0.017 ± 0.003 and

*d*

_{1}= −0.012 ± 0.004.

*τ*

_{υ}can be related to

*D*

_{m}(Figs. 4a and 4b) fromwhere

*D*

_{m}

*e*

_{0}and

*e*

_{1}/

*e*

_{0}for the midlatitude observations are 0.0191 and 0.0266 (correlation coefficient of curve fit is 0.89), respectively, and for the tropical clouds are 0.0166 and 0.0167 (correlation coefficient of 0.86), respectively.

A similar analysis of the relationship between *τ*_{υ} and *σ* was conducted by Matrosov and Snider (1995) using remote sensing data. They based their analysis on vertically pointing Doppler radar reflectivity and fall velocity measurements, coupled with infrared radiometer measurements, to obtain *τ*_{υ}, layer-averaged IWC, and *D*_{m}, using the radar radiometer approach (e.g., Matrosov et al. 1998). They found that for eight midlatitude cirrus cloud cases, *e*_{0} (cgs) ranged from 0.016 to 0.024, and *e*_{1}/*e*_{0} ranged from 0.018 to 0.024. The agreement between our midlatitude observations and those from the radar radiometer approach is remarkable and supports the overall validity of the two different approaches.

*τ*

_{υ}can be related to

*λ*(Figs. 5a and 5b) fromwhere

*λ*

^{−1}) is the layer-averaged slope of the gamma distributions fitted to the size distributions and the coefficients

*f*

_{0}and

*f*

_{1}/

*f*

_{0}for the midlatitude observations, which are 0.022 and 0.0044 (correlation coefficient of curve fit is 0.83), respectively, and for the tropical clouds, which are 0.0184 and 0.0043 (correlation coefficient of 0.94), respectively.

## Discussion

Properties of the particle size distributions are temperature dependent. As shown in Ryan (2000), Heymsfield et al. (2002a), and other studies, the slopes of exponential and gamma distributions fitted to PSD are strongly correlated with temperature. The monotonically decreasing values of *K* with temperature, shown in Fig. 2, suggest that the trends are related to the spectral slope. Further support for this view is suggested by the data collected during the loops of individual spirals, which have relatively constant values of *K* with small changes in temperature. An examination of the relationship between *K* and *λ* is especially important because *K* and *r*_{ge} are inversely related (Figs. 4c and 4d).

Heymsfield et al. (2002a) have developed analytic expressions for the extinction coefficient and ice water content in terms of the intercept *N*_{0}, slope *λ,* and dispersion *μ* of the gamma distributions, fitted to the particle size distributions. The term *K* = *σ*/IWC from the analytic expressions can be divided into three terms, as shown in the first three lines of Table 2. The first term (second line) depends primarily upon the habit. Because *k* is a constant and *a* is proportional to *λ,* term 1 is a function only of *λ* (see bottom of table). The second term (second line) depends on the ratio of two terms that depend primarily upon the order of the gamma distribution function and habit. The third term depends upon *λ* to a power related to the particle habit. As shown in the second line, *σ* and IWC are each dependent on *N*_{0} but *K* is not. Therefore, whereas IWC or *σ* fluctuates substantially during the individual loops of the spirals, *K* remains relatively constant. At the bottom of Table 2, each of the three terms is represented in terms of *λ,* and, therefore, *K* can be represented in terms of *λ.* The dependencies of *σ* as a function of *λ,* using the equations in Table 2, are plotted for the midlatitude and tropical datasets in Figs. 5a and 5b, where it is noted that they fit the data reasonably well. Because *λ* can be represented in terms of the temperature (see, e.g., HO3 for gamma distributions; Platt 1997 for exponential distributions), *K* is largely a function of temperature, a result that was shown in Fig. 2c.

Statistics characterizing the values of *D*_{m} and *r*_{ge} or *D*_{ge} in ice clouds, as a function of cloud type, temperature, and geographical location, are needed if the representations of *τ*_{υ} in terms of IWP and *r*_{ge} or *D*_{m} are to be useful for climate model studies, as well as for retrievals of *τ*_{υ} from measurements of IWP using conventional (non-Doppler, e.g., spaceborne) radars. Wang and Sassen (2002) describe a technique to estimate ice water path and layer mean *D*_{ge} using measurements of the optical depth as deduced from lidar measurements and the mean radar reflectivity factor of the cloud layer. This method is applicable when the optical depths are less than about 2 or 3 because a lidar beam is strongly attenuated for optical depths above these values. At higher *τ*_{υ} values, as was the case for most of our observations (Fig. 7a), it is desirable to use another method to derive values of *D*_{ge} or *D*_{m}.

The reflectivity-weighted mean terminal velocity (*V*_{Z}*D*_{m} and *r*_{ge} or *D*_{ge} (Matrosov et al. 2002; Mace et al. 2002). Each of the parameters *V*_{Z}*D*_{m}, and *r*_{ge} or *D*_{ge} are independent of the intercept parameter *N*_{0} of exponential and gamma size distributions, or for nonparameterized size distributions, independent of the relationship of concentration to size. Because terminal velocity and size are often related for given particle habits, there may be direct relationships between *V*_{Z}*D*_{m} or *D*_{ge} values that exhibit relatively little scatter.

HO3 described a method for deriving the terminal velocities of ice particle populations as a function of size and habit. The method draws upon recent drag coefficient data for various particle habits, including aggregates. Using this technique and the parameterized forms of the size distributions that explicitly contain small particles, it is possible to derive *V*_{Z}*V*_{Z}

In Figs. 8a and 8b, we calculated *V*_{Z}*V*_{Z}*D*_{m} and *r*_{ge}. The tropical observations show a strong Mie effect, especially at the smallest wavelengths, because of the presence of large particles. Power-law relationships fitted to the data and plotted in the figures can be used to derive *D*_{m} from suitably averaged *V*_{Z}*Z*_{e} at a frequency of 35 GHz, and are about 5 dB*Z*_{e} lower at a frequency of 94 GHz, with some of the replicator observations extending down to −45 dB*Z*_{e}. For the tropical observations, the radar reflectivities are primarily between −5 and 25 dB*Z*_{e}, at a frequency of 35 GHz, and about 5 dB*Z*_{e} lower at a frequency of 94 GHz. For *D*_{m} < 1 mm, the calculated values of *V*_{Z}

## Summary and conclusions

In this study, we have examined particle size distribution and particle area information obtained from Lagrangian spiral descents and balloon ascents through ice cloud layers, with an emphasis on examining the relationship between the ice cloud optical depth in visible wavelengths, and the ice water path. Although data collected in this way does not represent true vertical profiles, because of horizontal variability across the 5 km or so loops of the spirals, it did provide information on the vertical structure of the ice cloud layers, and the relationship between the optical depth in visible wavelengths and the ice water path. Clearly, because our dataset is small, comprising only 13 midlatitude and 6 tropical cases, and the temperatures of our observations may not be representative of ice clouds in an average sense, more in situ data of this type is needed. Despite these limitations, we were able to develop useful relationships between *τ*_{υ} and IWP, and an understanding of the factors that produce these relationships.

The relationships that we developed add to the work of Ebert and Curry (1992), Fu (1996), and Wang and Sassen (2002), who modeled ice cloud particles as hexagonal columns. Provided that the effective radius is defined on the basis of the representations given in those studies, the earlier relationships produce results that are consistent with our observations. Compared to these earlier studies, our study has a better definition of particle cross-sectional areas, better estimates and, for a limited dataset, direct measurements of the IWC, and size distribution measurements extending to much larger sizes. The use of Lagrangian vertical profiles through the depth of the ice cloud layers studied, the sampling of a wider range of temperatures, and the inclusion of both midlatitude and tropical clouds geographical locations allowed us to develop relationships between *τ*_{υ} and IWP that were functions of cloud layer–averaged properties. It was found that *τ*_{υ} and IWP could be related directly, or through a third parameter, including the cloud thickness, midcloud temperature, slope of the size distribution, median mass diameter, or effective radius. It is important to assess whether the direct relationship we found between *τ*_{υ} and IWP, without any intervening parameters, is applicable to ice clouds in general.

The relationships found between *τ*_{υ} and IWP in terms of temperature, cloud thickness, and cloud microphysical properties can be accounted for on the basis of the properties of the size distributions. At low temperatures and where ice clouds are optically thin, the extinction coefficient, which is dominated by the small particles, is large compared to the ice water content, which is dominated by the larger particles. The values of *K* in these regions were found to be approximately 0.06 g m^{−2}. Toward the base of geometrically thick ice clouds, where the temperatures are higher than at cloud top, and where particle size distributions tend to be broad and IWC increases more rapidly than does extinction, the values of *K* decreased to approximately 0.02 g m^{−2}. The nearly monotonic decrease in the values of *K* downward from cloud top was shown both empirically and analytically to result from the slope of the size distribution, which tends to flatten with increasing cloud depth and increasing temperature. Number concentration has a minor influence on the relationship between *τ*_{υ} and IWP, because both *σ* and IWC are affected equally by the concentration, assuming that either exponential or gamma distributions fit the actual size distributions reasonably well.

Our results are found to correspond closely with relationships derived between *τ*_{υ}, IWP, and *D*_{m} from retrievals using a combination of vertically pointing Doppler radar and infrared radiometer data by Matrosov and Snider (1995). It may be possible to improve upon such estimates in the future through the use of the relationships we developed to retrieve *D*_{m} and *D*_{e}, which are proportional to 1/(*K*), from vertically pointing Doppler radar data. Additional in situ data are clearly needed in a variety of geographical locations to refine the relationships developed here.

The authors thank Aaron Bansemer and Larry Miloshevich for their scientific input and Beverly Armstrong for her editorial assistance. We thank Greg McFarquhar for his help with the CEPEX parameterization and data, and two anonymous reviewers for their detailed reviews. This research was supported by the NASA grant number L-16441, sponsored by Dr. Donald Anderson, and by NASA EOS Grant S-97894-F.

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# APPENDIX A

## List of Symbols

- Symbol Description
*A*Ice particle cross-sectional area*A*_{c}Total cross-sectional area of particle population per unit volume (cm^{−3})*A*_{r}Area ratio—particle area divided by area of circle with the same*D**a,**b*Coefficient, exponent in the fit of the area ratio–vs-diameter data from 1-km imaging probe data*c*_{0},*c*_{1}Coefficients in relationship between*K*and height above cloud base*D*Particle dimension as imaged by 2D probes from diameter of circumscribed circle*D*_{e}Effective diameter—diameter used to characterize the radiative properties of a particle population*D*_{ge}Fu (1996)'s definition of effective diameter—diameter used to characterize the radiative properties of a particle population*D*_{m}, Median mass diameter, layer-averaged median mass diameter*D*_{m}*D*_{max}Maximum measured particle size in a given 7-s sample*D*_{melt}Melted equivalent spherical diameter*d*_{0},*d*_{1}Coefficients in*K*relationship with temperature*e*_{0},*e*_{1}Coefficients in relationship between*K*and*D*_{m}*f*_{0},*f*_{1}Coefficients in relationship between*K*and*λ**g*_{0},*g*_{1}Coefficients in relationship between*K*and*D*_{e}*h,**h*_{base},*h*_{top}, Δ*h*Height (altitude), cloud-base and-top heights, cloud thickness (km)- IWC Ice water content
- IWP Ice water path, the integral of IWC through cloud depth
*K*Mass extinction coefficient, ratio*σ*/IWC*k*Coefficient in equation for effective density*m*Ice particle mass*N*Concentration per unit diameter as a function of*D*(cm^{−4})*N*_{0}Concentration intercept parameter*n*Exponent in effective density relationship*r*_{e}, Effective radius, layer-averaged effective radius*r*_{e}*r*_{ge}, Fu (1996)'s definition of effective radius (Fu 1996), layer-averaged effective radius*r*_{ge}*T*Temperature (°C)*V*_{Z}, Reflectivity-weighted terminal velocity, mean terminal velocity*V*_{Z}*α*Exponent of*D*in effective density relationship*β*Coefficient in mass-vs-diameter relationship*λ*Slope parameter of gamma size distribution*μ*Dispersion of gamma particle size distribution*ρ*_{e},*ρ*_{i}Effective ice density, solid ice density*σ*Extinction coefficient (m^{−1})*τ*_{υ}Optical depth in visible wavelengths, the integral of*σ*through cloud depth*χ*Exponent in mass-vs-diameter relationship

# APPENDIX B

## Small Particle Contributions to Tropical Mass Extinction Coefficients

McFarquhar and Heymsfield (1997) developed a parameterization for the ice-crystal size distributions in terms of the melted diameters of the particles for tropical ice clouds observed during the Central Equatorial Pacific Experiment (CEPEX). The parameterizations are based on data from a video ice particle sampler that measured the concentrations of particles down to about 10-*μ*m diameter, and a 2D-C probe that sized particles from about 50 *μ*m to about 1000 and 2000 *μ*m. Given a value of the IWC (between 0 and 1 g m^{−3}), as measured by a 2D-C probe and a temperature (between −70° and −20°C), the McFarquhar and Heymsfield parameterization returns a size distribution of melted equivalent diameter spheres that is the sum of a first-order gamma function, describing ice crystals with melted equivalent diameters less than 100 *μ*m, and a lognormal function, describing larger ice crystals. Here, we will use our tropical IWC values below −20°C and the corresponding temperature measurements together with the CEPEX parameterization to obtain estimates of the total IWC that includes the portion in sizes (of melted diameter, *D*_{melt}) below 40 *μ*m, the total *σ* that the parameterization returns, and the *σ* that the 2D-C probe would have seen. The latter value is not what we observed but is consistent with the parameterized size distributions. Note that it was not possible to perform these calculations for all cases, because sampling for several of the tropical cases was conducted at temperatures above −20°C.

Figure B1a shows the relationship between *σ* and IWC for sizes (*D*_{melt}) above 40 *μ*m (gray symbols), approximately what the 2D-C probe would have seen, to the corresponding values of these parameters in all sizes (black symbols). It is noted that the omission of small particles may lead to a distribution of *σ* versus IWC relationship that is monotonic on a log–log plot.

Figure B1b shows the ratios of the values of *K* for all sizes to those where *D*_{melt} > 40 *μ*m, as a function of *K* for all sizes. The average value of these ratios of *K* is 1.06 ± 0.09 for the tropical size distributions, and 1.30 ± 0.22 for the midlatitude distributions. The averages are shown with horizontal lines in Fig. B1b.

For each Lagrangian spiral descent or balloon ascent, the values of IWC and *σ* from the parameterization are integrated downward through the cloud layer (where the temperatures were below −20°C) to estimate how much larger the actual cloud optical depth and ice water path might have been, relative to those that were observed. The replicator observations are included in these estimates, as though the size distributions are measured with a 2D-C probe only, by using the portion of the size distributions above 40 *μ*m. Figure B1c shows that the enhancement in the *τ*_{υ} values through the layer are larger than the enhancement in the IWP values. The enhancements for the tropical datasets, those containing the largest *τ*_{υ} values, are modest, consistent with the results shown in Fig. B1b. The layer-averaged enhancements for the replicator cases are much larger than are observed (shown previously in Figs. 2a and 2b).

Average enhancements in the values of *τ*_{υ} relative to IWP are shown in Fig. B1d as a ratio and are represented as *K.* It is noted that the average enhancement increases with decreasing *τ*_{υ} values, and are modest for the tropical clouds. The layer-averaged enhancement for the replicator cases are much larger than observed (Fig. 2C), therefore, it is concluded that the estimates in Fig. B1, as a whole, overestimate the effects of small particles.

Data sources for vertical profiles through cloud depth

^{−4}is for units conversion,

*a*and

*b*are the terms in the area ratio relationship

*A*=

_{r}*aD*, and the total area of the ice particle population is

^{ b}*A*= (

_{c}*π*/−4)

*aD*

^{ 2+b}. The coefficients

*a*and

*b*are represented in terms of λ in H03. The coefficents

*k, n,*and α are terms in the expression for ice particle density given in Heymsfield et al. (2002b). There is a correlation between λ and μ. There is also a direct relationship between term 1 and λ, because for term 1,

*k*is a constant; and in the parameterization

*a*is represented in terms of λ

^{1}

Detailed discussions of the role of small particles in tropical ice clouds during the Central Equatorial Pacific Experiment (CEPEX), as measured using a video ice particle sampler and 2D imaging probes, is given in Heymsfield and McFarquhar (1996). When IWC < 1.0 × 10^{−4} g m^{−3}, 100% of the IWC was in sub-90-*μ*m particles; at IWC = 3.0 × 10^{−3} g m^{−3}, this fraction decreased to 50%; and for IWC = 0.01 g m^{−3}, this fraction decreased to <20%. Almost all of *σ* was measured when *σ* > 1.0 × 10^{−4} or 2.0 × 10^{−4} m^{−1}. These values are consistent with the replicator observations shown in Fig. 2, and are supported, for the most part, by the observations in midlatitude ice clouds by Arnott et al. (1994, their Fig. 8c). A value of *σ* above 2.0 × 10^{−4} m^{−1} was far exceeded for all of our TRMM cases and for all but the very upper parts of the spirals for FIRE I. The value was far exceeded throughout most of the ARM spirals.