## Abstract

Knowledge of ice crystal particle size distributions (PSDs) is critical for parameterization schemes for atmospheric models and remote sensing retrieval schemes. Two-dimensional in situ images captured by cloud imaging probes are widely used to derive PSDs in term of maximum particle dimension (). In this study, different definitions of for nonspherical particles recorded by 2D probes are compared. It is shown that the derived PSDs can differ by up to a factor of 6 for *μ*m and mm. The large differences for *μ*m are caused by the strong dependence of sample volume on particle size, whereas differences for mm are caused by the small number of particles detected. Derived bulk properties can also vary depending on the definitions of because of discrepancies in the definition of used to characterize the PSDs and that used to describe the properties of individual ice crystals. For example, the mass-weighted mean diameter can vary by 2 times, the ice water content (IWC) by 3 times, and the mass-weighted terminal velocity by 6 times. Therefore, a consistent definition of should be used for all data and single-particle properties. As an invariant measure with respect to the orientation of particles in the imaging plane for 2D probes, the diameter of the smallest circle enclosing the particle () is recommended as the optimal definition of . If the 3D structure of a particle is observed, then the technique can be extended to determine the minimum enclosing sphere.

## 1. Introduction

Ice clouds play important roles in the atmosphere through latent heat release and radiative transfer, which are determined by the underlying microphysical processes. Cirrus, the most common form of ice clouds, covers around 20% of the earth, and hence its influence on radiation is essential for the earth’s energy balance (Heymsfield and McFarquhar 2002). To understand the properties of and processes occurring in ice clouds, realistic particle size distributions (PSDs) and bulk properties of ice clouds are needed and are typically obtained from in situ observations. Assumptions about the form of PSDs based on the in situ observations are then made in parameterization schemes that are used in atmospheric models and remote sensing retrievals.

The PSDs are typically derived from two-dimensional images obtained in situ by probes installed on aircraft flying through clouds and by disdrometers on the ground. Two-dimensional optical array probes, which give such images of cloud and precipitation particles, were originally developed by Knollenberg (1970). Different versions of such probes are now available, such as Particle Measuring Systems’s (PMS) 2D cloud (2D-C) and 2D precipitation (2D-P) probes (Knollenberg 1981), Droplet Measurement Technologies’s (DMT) cloud imaging probe (CIP) and precipitation imaging probe (PIP) (Baumgardner et al. 2001), and the Stratton Park Engineering Company’s (SPEC) two-dimensional stereo (2D-S) probe (Lawson et al. 2006) and high-volume precipitation spectrometer (HVPS) (Lawson et al. 1993). These probes provide information on the sizes, shapes, and projected areas of ice particles with dimensions greater than about 10 *μ*m, with the exact size range of each probe depending on the magnification, resolution, and distance between probe arms. High-resolution (2.3 *μ*m) images of cloud particles can also be obtained from a charge-coupled device (CCD) camera on SPEC’s cloud particle imager (CPI) (Lawson et al. 2001). Further, holographic images can be constructed from two-dimensional images obtained by the Holographic Detector for Clouds (HOLODEC) (Fugal et al. 2004). Various kinds of disdrometers, such as the 2D video disdrometer (2DVD) (Kruger and Krajewski 2002) and the snow video imager (SVI) (Newman et al. 2009), also obtain two-dimensional images of particles at the ground.

From observed PSDs, various bulk cloud properties, such as total number concentration, extinction, liquid and ice water content, mean fall speed, precipitation rate, effective diameter, and single-scattering properties can be determined. However, calculation of these parameters is complicated by the fact that different definitions of particle size have been used to characterize PSDs and the functional relationships between single particle properties and the particle size, and these definitions of particle size are not always consistent with definitions used to describe the properties of individual crystals. For example, even though many studies use the maximum diameter () as a measure of particle dimension (Locatelli and Hobbs 1974; McFarquhar and Heymsfield 1998; Petty and Huang 2011; Jackson et al. 2014; Heymsfield et al. 2013; McFarquhar and Heymsfield 1996; Mitchell and Arnott 1994; McFarquhar and Black 2004; Baran et al. 2014; Korolev et al. 2014; Korolev and Field 2015), area-equivalent diameter () (Locatelli and Hobbs 1974; Korolev et al. 2014), and mass-equivalent diameter (or melted diameter ) (Seifert and Beheng 2006) have also been used (Table 1). Although all definitions are equivalent for spherical liquid particles, there can be large differences for nonspherical ice particles, as has been noted for optical array probes (OAPs; Brenguier et al. 2013), imaging disdrometers (Wood et al. 2013), and nonimaging disdrometers (Battaglia et al. 2010). This has important ramifications. For example, McFarquhar and Black (2004) noted that inconsistencies in particle size definitions could have significant impacts on mass conversion rates between different hydrometeor classes used in numerical models. Consistency in the definition used to characterize the PSDs and libraries of particle properties (e.g., mass or scattering properties) is needed to compute bulk or optical parameters.

Even if maximum dimension is used to represent PSDs, there are different methods that have been used to calculate . Depending upon the definition of maximum dimension used in the library of particle properties, these differences can be problematic. The should be the longest dimension in any direction across the 3D volume of the particle, which is equivalent to the diameter of the smallest sphere enclosing the particle. However, because of the limits of measurement technologies, only two-dimensional projections of a particle in one specific orientation can be observed by 2D imaging probes. Therefore, the true maximum dimension of a particle is not directly available, and it can be only estimated from the 2D projected images until new technologies that provide the 3D structure of a single particle emerge. In this study, the focus is on the calculation of a maximum dimension describing the two-dimensional projection of a particle and the ramifications of differences in these definitions. This extends previous studies that examined how the calculation of affected properties derived by disdrometers (Wood et al. 2013) and that identified how differences in definitions could impact PSDs from OAPs (Brenguier et al. 2013).

There are several different ways has been calculated for a two-dimensional image (Locatelli and Hobbs 1974; Brown and Francis 1995; McFarquhar and Heymsfield 1996; Mitchell and Arnott 1994; Korolev and Field 2015; Heymsfield et al. 2013). In this study, the impacts of different definitions of on PSDs and bulk cloud properties are explored. Differences in bulk properties between the various definitions of are determined, as are differences in such properties using consistent and inconsistent definitions of in the derived PSDs and libraries of microphysical and scattering properties.

The methods for calculating the for 2D imaging probes are described in section 2. Section 3 describes the field campaigns and probes from which the in situ data were acquired. The differences in the derived PSDs from one flight are discussed in section 4, and the differences in bulk cloud parameters calculated from the PSDs, such as ice water content (IWC), mass-weighted terminal velocity, precipitation rate, extinction, and effective radius, are discussed in section 5. Section 6 summarizes the findings and their significance.

## 2. Definitions of *D*_{max}

*D*

_{max}

In this section, different methods of determining from two-dimensional particle images are discussed. In addition, a new method for computing as the diameter of the smallest enclosing circle () is described. To understand how the maximum dimension of a nonspherical ice particle is defined, it is helpful to first review how cloud imaging probes work. This is done in the context of OAPs since they are commonly used for measuring PSDs, but the results apply to any probe acquiring a two-dimensional image. OAPs consist of an array of photodiode detectors that record light emitted from a laser beam. When a particle passes through the laser beam, a number of diodes proportional to the size of the particle are shadowed along the direction of the photodiode array (Brenguier et al. 2013). Because the diodes are clocked by fast response electronics at a rate proportional to the width of photodiode element, the particle is also measured along the direction of aircraft flight (time direction). Thus, a two-dimensional image is recorded. These two directions are shown in Fig. 1 and are designated for the maximum dimension in the time direction and for the maximum dimension in the photodiode array direction.

In the past, at least five different definitions of have been commonly used. The maximum particle dimensions previously used include the maximum dimension in the time direction (), maximum dimension in the photodiode array direction (), the larger of and () (McFarquhar and Heymsfield 1996), the mean of and () (Brown and Francis 1995), and the length of the hypotenuse of a right-angled triangle () constructed from the two dimensions and calculated as (Mitchell and Arnott 1994). Figure 1 illustrates the four definitions of (excluding and for the particles depicted) for an ice particle example imaged by the HVPS probe during the Midlatitude Continental Convective Cloud Experiment (MC3E).

A new algorithm for determining the maximum dimension of a two-dimensional projected image of a particle as the diameter of a minimum enclosing circle is also used in this study. The problem of finding a minimum enclosing circle of a nonspherical particle is a classical computational geometry problem, and solutions are readily available. The origin of such a circle is the perfect location for a public service, such as a hospital or a post office, because it minimizes the distance from the service for all residents (De Berg et al. 2008). The more general problem of finding a minimally enclosing sphere in *N* dimensions is the so-called Euclidean 1-center problem (Gärtner 1999). There have been many efforts to derive a fast algorithm to determine the smallest enclosing *N*-spheres, with time complexity ranging from *O*(*n*^{4}) to *O*(*n*). Because of the large number of ice crystals that are typically measured during a flight, it is important to implement the fastest possible algorithm in probe processing software. Historically, the optimal algorithm was thought to have time complexity of *O*[*n* log(*n*)] (Shamos and Hoey 1975), until the first linear-time algorithm was proposed by Megiddo (1982) using a linear programming method. More recently, Welzl (1991) developed a simple randomized linear-time algorithm, with a subsequent implementation developed by Gärtner (1999). This algorithm employs a stochastic method to rapidly determine the minimum surface for dimensions less than 10. This algorithm has been adopted in the University of Illinois at Urbana–Champaign optical array probe software for the two-dimensional cloud particle images and is used to compute in this study.

## 3. Dataset

To test the newly implemented Gärtner (1999) algorithm for calculating , in situ measurements acquired by airborne probes during MC3E (Petersen and Jensen 2012), jointly sponsored by the National Aeronautics and Space Administration (NASA) and the U.S. Department of Energy (DOE) Atmospheric Radiation Measurement (ARM), are used. MC3E was conducted in April and May of 2011 in the vicinity of the DOE-ARM Southern Great Plains (SGP) Climate Research Facility in northern Oklahoma. During the field campaign, the University of North Dakota (UND) Citation sampled clouds in 12 weather systems, including fronts, squall lines, and MCSs.

For this study, the MCS that passed over the ARM SGP site from the west on 20 May 2011 was chosen for analysis because all of the in situ probes worked well. On this day, a deep trough in upper levels was collocated with the lower-level jet stream, providing a favorable synoptic setting for the development of convection. At the same time, the lower-level jet stream supplied a large amount of moisture from the Gulf of Mexico to fuel the convection. Vertical wind shear was also present, which allowed the MCS to persist for a longer time period compared to the low shear condition. During the 4-h flight, which started at 1255 central daylight time (CDT), from Ponca City, Oklahoma, the UND Citation sampled the stratiform region behind the convective line. As shown in Fig. 2, the UND Citation executed several constant-altitude stepped legs, and one upward and one downward spiral over the SGP. The UND Citation ascended as high as 7.6 km and sampled clouds with temperatures ranging from −23° to 20°C. In this study, only data in ice clouds are used.

A variety of particle habits was sampled during the flight. Figure 3 shows representative particles as a function of temperature imaged by the 2D-C. Most particles were classified as “irregular” by a habit identification routine (Holroyd 1987). The roundish shape of many of the particles suggests that they might have experienced some riming during their growth history. But, since there was little or no liquid water content measured at temperatures below 0°C during the flight, their masses and areas were calculated using Brown and Francis (1995) mass- and area-dimensional relationships that were derived for midlatitude cirrus that also consisted of predominantly quasi-spherical irregular particles, with some bullet rosettes and columns mixed in.

The CIP, 2D-C, and HVPS were installed on the UND Citation, and nominally sampled particles between 25 *μ*m and 19.2 mm. In this study, data from the 2D-C and HVPS are combined to give a composite PSD. The 2D-C is used to characterize particles smaller than 1 mm, while the HVPS is used for sizes larger than 1 mm. As is shown in Fig. 4, the 1-mm cutoff was chosen since it is around the center of the size range where for the 2D-C and HVPS agree within 5% for this flight. It should be noted that a small and poorly known depth of field for particles with 150 *μ*m can cause a substantial uncertainty in for *μ*m (Heymsfield 1985; Baumgardner and Korolev 1997). In this study, the smallest bin is set to be 150 *μ*m to eliminate this uncertainty. Further, given the 30-*μ*m resolution of the 2D-C, any particles with *μ*m would have at most five photodiodes shadowed, meaning there would be poor resolution for looking at the effects of particle shape in the computation of .

The 2D-C was used for the analysis instead of the CIP because the 2D-C was equipped with antishattering tips, while the CIP was not. Large numbers of small ice crystals can be produced when a large crystal shatters on the tips of an OAP; therefore, antishattering tips have been developed to deflect such shattered particles away from the probe sample volume (Korolev et al. 2011). Korolev et al. (2011, 2013a) and Jackson et al. (2014) have shown that some particles with *μ*m are shattered artifacts, even when antishattering tips are used. Shattered artifacts are identified as those particles with interarrival times below some threshold. Typically there are two peaks in the interarrival times, with the smaller peak corresponding to the shattered remnants and the larger peak corresponding to real particles (Field et al. 2006; Korolev and Field 2015). However, the time evolution of the frequency distribution of interarrival times in Fig. 5 illustrated only a single mode in the distribution of interarrival times for the 2D-C and HVPS. Therefore, there is no peak in the interarrival time analysis corresponding to shorter times, suggesting few artifacts were generated by the shattering of large crystals on the probe tips for conditions sampled during this flight. Therefore, no shattering removal algorithm was used for both the 2D-C and HVPS.

The University of Illinois software determines various measures of particle morphology, such as particle dimension, projected area, particle habit, particle mass, rejection status, area ratio, and interarrival times. The code is modified to include the calculation of and other definitions of so that alternate versions of PSDs were generated. The projected area of a single particle can be directly determined if the particle image is entirely within the photodiode array. However, many particles touch the edges of the photodiode array and therefore additional assumptions are needed to estimate the projected area. To get the projected area for particles touching the edge of the photodiode array, various area-dimensional (*A*-*D*) relations can be used to calculate the single-particle projected area, where *D* is the reconstructed dimension of the particle (Heymsfield and Parrish 1978). In this study, the projected area for each particle is determined using *A*-*D* relations as well as the directly imaged area. Traditionally, *A*-*D* relations are represented by power laws, where

with *a* and *b* as habit-dependent parameters. Particle mass (*m*) is not observed by the imaging probes. To get the particle mass, mass-dimensional (*m*-*D*) relations are also assumed and are again represented by power laws, such as

where *α* and *β* are habit-dependent parameters. In this study, Holroyd’s (1987) habit classification is used to determine ice particle habits. Subsequently, the appropriate *A*-*D* and *m*-*D* relations from Mitchell (1996) are used to give the habit-dependent *a*, *b*, *α*, and *β* parameters listed in Table 2.

To calculate the PSDs, an assumption must be made about the probe sample volume, since the number distribution function is calculated as the observed number of counts in each bin divided by the sample volume and bin width. The sample volume is calculated using

where DOF is depth of field, is the distance between the probe arms, is the effective width of the photodiode array, and TAS is the true airspeed. The DOF is calculated as

for the 2D-C and CIP (Knollenberg 1970), where *λ* is the wavelength of laser. The use of to calculate the DOF has been suggested (A. Korolev 2015, personal communication) since the OAPs take only one slice of a particle at a time, but there is currently no consensus about which definition of size should be used for calculating the DOF for irregular ice particles (Brenguier et al. 2013). This uncertainty contributes to the uncertainty in the probe sample volume. Another source of uncertainty comes from the determination of . There are three ways to calculate : entire-in, center-in, and the Heymsfield and Parrish (1978) extension. Figure 6 shows the sample volumes calculated using these three methods for the 2D-C, CIP, and HVPS. These three methods are different in their treatment of partially imaged particles. The entire-in technique uses only the fully imaged particles but reduces the sample volume linearly as the particle size increases. On the other hand, Heymsfield and Parrish (1978) utilizes all particles, calculating their size based on the assumption of spherical particles, and thus extends the sample volume but having more uncertainty in the estimated particle area. The center-in technique represents a trade-off between the volume of data and the quality of the images. This method uses partially imaged particles whose center is inside the sample volume, in addition to the fully imaged particles. A particle is determined to be a center-in particle when its maximum dimension in the time direction does not touch the edge of photodiode array. There is a smaller uncertainty in the imaged area for a center-in particle because a greater fraction of the particle was imaged. In this study, the center-in technique is used. In addition to the number distribution function , the area size distribution functions and the mass distribution function are also derived.

The bulk cloud properties are related to specific moments of the PSDs or are obtained by integrating the area or mass distribution functions. For example, the total number concentration is calculated as

and the ice water content is given by

The Nevzorov total water content (TWC) sensor was also installed on the UND Citation to measure both liquid and ice water content (Korolev et al. 1998), which can be used to constrain and validate the assumptions used for calculating IWC from OAPs. For ice clouds, the TWC is the same as the IWC. The mass flux, or precipitation rate (PR), is expressed using

where is the terminal velocity of an individual particle calculated from particle area, mass, temperature, and pressure following Heymsfield and Westbrook (2010). Following McFarquhar and Black (2004), the mass-weighted terminal velocity is expressed as

The extinction () at visible wavelength is twice the total projected area () since the ice particles are large enough that geometric optics applies (Um and McFarquhar 2015). The is the integrated projected area of particles over all sizes, given by

The effective diameter () is commonly used for parameterization of single scattering properties needed for calculation of shortwave radiative transfer (Fu 1996; McFarquhar and Heymsfield 1998; Mitchell 2002). Although several different definitions of have been used (McFarquhar and Heymsfield 1998), is defined here following Fu (1996) as

because the ratio of to is closely related to ice radiative properties. In Eq. (11), is the bulk density of ice, assumed to be 0.91 g cm^{−3}. Number-weighted mean dimension (), or the average particle dimension, is defined as

whereas the mass-weighted mean diameter () is calculated as

Besides the bulk properties addressed here, there are also differences in the radar reflectivity derived from PSDs using alternate definitions of . In the Rayleigh scattering regime, the radar reflectivity is a higher-order moment of the PSDs than the quantities discussed here (Smith 1984). However, different models exist for the calculation of radar reflectivity from ice particles, and consideration of all these different models is beyond the scope of this study.

For calculating quantities in Eqs. (5)–(12), the measured particles are first sorted into bins of varying width with ranging from 150 *μ*m to 1.92 cm. Then the integrations are converted to summations, with the relevant particle properties computed at the midpoint of the bin. The midpoint rule gives a better estimate of integration for the concave-down shape of the PSDs compared with the trapezoidal rule. Given that the number of bins is not large, more complex numerical methods (e.g., Simpson’s rule) may not be needed. Based on these calculations, the impacts of different definitions of on PSDs and bulk cloud properties are explored in the next two sections.

## 4. Effect of definitions on PSDs

The composite PSDs computed using six different definitions of for the MCS on 20 May 2011 are compared in this section. The were first determined for each 1 s of flight time, and averaged PSDs were subsequently computed in three different temperature ranges. Figure 7 compares the determined using the six different definitions, with the upper panels showing and the lower panels showing the ratio of to , with the ratio being one when is used as . The all show a peak between 200 and 400 *μ*m for all temperature ranges. For 300 *μ*m, decrease sharply to eight orders of magnitude smaller, with the rate of decrease depending on temperature. The using the different definitions of can vary by up to one order of magnitude with, for example, and at ~1 mm between −10° and 0°C varying by this amount.

The trends in how the different definitions of vary are systematic in that , , , and are always smaller than or equal to , while is always greater than or equal to , as shown for the example particle in Fig. 1. Consequently, , , , and are larger than for smaller and smaller for larger . The trend for compared to is opposite. For all the definitions of , is the closest to . The differences between using different definitions of increase when is farther away from the mode diameter, for both smaller and larger sizes.

Since the number distribution function is determined by the number of counts in each bin and by the sample volume for particles with the given size, both factors contribute to the differences in the PSDs. For *μ*m, the large difference between PSDs is due to the dependence of the depth of field, and therefore the sample volume, on particle size. This increases if the particles are moved from a larger bin to a smaller bin because of the different definitions of . The effect is larger as the particle size, and therefore the DOF and the sample volume, decreases. For *μ*m, the sample area is constant (red solid lines in Fig. 6), so that the changes in PSDs for the different definitions are due to the number of particles sorted into each bin, which is determined by the definition of . The PSDs are more sensitive to the choice of definitions for larger particles than for smaller particles because the number of particle counts per bin decreases sharply as the particle size increases; thus, the classification of even a single particle into a different bin can have a big impact.

When comparing the behavior of PSDs for different temperatures in Fig. 7, it is apparent that the differences in for different definitions are larger for lower temperatures than for higher temperatures. This might be explained by more circular particles at higher temperatures due to the action of riming and aggregation. It is also noticeable that the slope of the PSDs increases for the lower temperatures, meaning fewer large particles exist when the temperature is lower. This would again lead to larger differences in the PSDs.

The impact of the different definitions of is also seen when comparing the number-weighted mean diameter () and computed from determined using different definitions of . Figure 8a compares calculated using five definitions of to that calculated using . The differences are also summarized in Table 3. The range from 300 to 1200 *μ*m, with the differences varying between 56% and 140% due to the different definitions of . Figure 8b shows the comparisons for . The differences in vary from 65% to 125% and the values range from 300 *μ*m to 8 mm. Using and gives the smallest and , with and about 74.5%–79.6% and 83.9%–87.5% of these computed with , respectively. The and determined using are similar, with a median ratio of 77.9% and 81.2% of those determined using , respectively. The and determined using provide the closest estimate to these determined using , with average differences of 91.7% and 93.3%, respectively. On the contrary, both and computed using are systematically larger than the and computed using , with up to a 140% difference.

Two factors contribute to the differences between and determined using different definitions of . First, the differences in are large as shown in Fig. 7. Since there are more smaller particles when using , , , and to define than when using , the and using these definitions are smaller than and calculated using , respectively. The second reason for the difference in the computed is that the assumed *m*-*D* relations are not applicable with certain definitions of . If the same coefficients in the *m*-*D* relations are used for different , then large differences in derived particle mass exist. For example, by using Brown and Francis (1995), gives the estimate of mass that is most consistent with the derivation from the original relationship, while and underestimate the particle mass and , , and overestimate the particle mass. It is also important to note that there is no consistent definition of used in different studies giving *m*-*D* relations. This point will be discussed in more detail in the subsequent section.

## 5. Effect of definitions on bulk properties

The differences in PSDs translate into differences in bulk properties. In this section, the influence of different definitions of on bulk cloud properties, such as , , , , , and , is examined.

### a. Total number concentration

The is obtained by integrating the number distribution function [Eq. (3)]. The definition of affects the derived as shown in Fig. 9a, with varying from 80% to 140% of that estimated using . Even though the same number of particles is recorded by the probe regardless of the definition of used, the changes with the definition of because of the dependence of the estimated probe DOF on . In general, definitions that give larger values of than for the same particle, such as , produce smaller , with values ranging from 94.7% to 99.8% of those obtained using within the 5th–95th percentiles, and a median of 98.0%. On the other hand, definitions that give smaller values of than , such as , , , and , have larger . For example, derived using and range from 101.8% to 126.0% and 104.0% to 119.5% within the 5th–95th percentiles of that derived using , and a median of 106.8% and 107.7%, respectively. The derived using and gives closer values to those derived using , with a median of 103.1% (99.9%–109.6% within the 5th–95th percentiles) and 101.3% (100.4%–103.3% within the 5th–95th percentiles) of those obtained with , respectively.

### b. Ice water content

The IWCs calculated using different definitions of and the TWC measured by the Nevzorov probe are shown in Fig. 9b as a function of the IWC calculated using . When the IWCs derived using different definitions of are compared with the TWC observed by Nevzorov probe, it is seen that the Nevzorov TWCs are less than the IWCs computed from when the IWCs get larger. This could be explained by the difficulties associated with the Nevzorov probe’s ability to sample larger ice particles (Korolev et al. 2013b). In addition, power-law fits are less likely to perform well at the extremes for the estimate of bulk properties. The IWCs determined using alternate definitions vary between 50% and 150% of those determined using . The IWCs calculated using give the largest estimate, ranging from 137.2% to 149.3% within the 5th–95th percentiles of those determined using with a median of 142.9%. In addition to factors leading to varying , differences of IWCs are caused by the use of different in the mass-dimensional relations, which are inconsistent with the definitions of originally used to develop the relations. The IWCs determined using are larger than those determined using other definitions since is the largest value of any and hence gives the largest estimated particle mass given the use of the same *α* and *β* coefficients in the *m*- relations. Defining as and produced the smallest IWCs because those particle dimensions are measured only in one direction, and hence they and their associated masses are smallest. The IWCs calculated using are closer to those calculated using than those calculated using (median ratio of 70.0% vs 56.9%). This occurs because the full dimension of particles that touch the edge of photodiode array is not recorded, but the longer dimension can be recorded in the time direction. For similar reasons, IWCs estimated based on are likely underestimated. The IWCs derived using are closest to IWCs derived using , with a median ratio of 81.9%.

For the implementation of an *m*- relation, it is important that the definition of used be consistent with the definition used in the relevant study that derived the relations. However, this is not always the case. For example, Brown and Francis (1995) used to calculate the mass of aggregates with the *m*-*D* relation that is originally documented by Locatelli and Hobbs (1974, p. 2188) for “aggregates of unrimed radiating assemblages of plates, side planes, bullets, and column.” In spite of the large differences, possible conversions between different definitions of may provide a way to correct the *m*- derived using different definitions. But, it is difficult and nontrivial to correct the *m*-*D* relations so that they apply to alternate definitions of because conversions between different definitions of depend upon morphological features of ice crystals that are not always reported in original studies. For example, Fig. 10 shows that there is large scatter between the different definitions of on a particle-by-particle basis, and no simple relations can be found between different definitions of . For the same particle with different orientations in the two-dimensional imaging plane, the is invariant; however, other definitions show wider scatter due to different orientations, especially for (Fig. 10a) and (Fig. 10b). The combination of and can reduce the scatter significantly, even though there are still systematic differences among different methods. As a result, it appears that methods that involve consideration of the particle dimension in at least two different directions (e.g., and ) are needed to get a reasonable estimate of the . This is similar to the findings of Wood et al. (2013) for the two-dimensional video disdrometer, which allows particle dimensions to be measured from two perpendicular views.

### c. Mass-weighted terminal velocity

Figure 9c shows that determined using varying definitions of can vary from 28% to 180% compared to those determined using . As with IWC, larger occur for definitions that give larger particle sizes, with median determined using at 134.0% (120.1%–166.1% within the 5th–95th percentiles) of those determined using . Similarly, the smallest were associated with the definitions giving smaller particle sizes, namely, and , with a median ratio of 54.6% (28.7%–82.0% within 5th–95th percentiles) and 74.5% (66.3%–85.4% within 5th–95th percentiles) of those determined using , respectively. In general, the differences are larger when m s^{−1}. When exceeds 0.6 m s^{−1}, the differences decrease to be within 60%–145%, and the computed using the other definitions converge to calculated using . The large spread in is contributed by both the variations in particle area and mass used for the calculation of , since both particle area and mass estimated using the power laws vary due to the different definitions of . The differences of the precipitation rate is also presented in Fig. 9d, which shows a similar pattern for since PR is the combined effects of and IWC, and the uncertainties in are much greater than that in IWC.

### d. Extinction

Figure 9e shows the variation in extinction determined using different definitions of and the directly imaged area determined from the OAPs. The extinction determined using different definitions of can vary from 60% to 133% of those calculated using . Similar to the result shown for IWC, determined using ranges from 126.7% to 133.4% of that determined using , with a median of 130.4%. The uncertainties in both and the *A*-*D* relations contribute to the differences. When using definitions of that produce smaller particle sizes, the values are smaller than those determined using with, for example, medians of 65.4% and 71.9% for and , respectively, compared to those determined using . Because the OAPs directly measure particle area, there is some measure of truth for particle area or, equivalently, extinction. Computations of using are closest to the estimated from the OAP directly imaged area. This is not surprising because many of the *m*- relations are based on the use of as the maximum diameter. The patterns of differences in extinction between definitions are quite similar to those seen in IWC. However, the differences are smaller, because is based on a lower-order moment compared to IWC.

### e. Effective diameter

Figure 9f shows the effective diameter calculated using the different definitions of compared to that determined using . The computed using the different definitions range from 82% to 120% of the computed using . Since is related to the ratio of IWC to , the dependence of both these variables on the definition of influences . In general, the computed using are largest, ranging from 106.9% to 113.7% of the calculated using within the 5th–95th percentiles, with a median of 109.7%. However, using definitions that give smaller particle sizes can also give higher estimates, especially for larger values of . When is less than 80 *μ*m, using , , and still give smaller estimates than those calculated using . The calculated using and produced the smallest estimates, with medians of 86.9% (82.3%–94.7% within the 5th–95th percentiles) and 96.7% (92.8%–105.7% within the 5th–95th percentiles) of those determined using , respectively. Using and give estimates of very close to those determined using , with medians of 88.9% (84.2%–96.1% within the 5th–95th percentiles) and 96.4% (94.9%–98.2% within the 5th–95th percentiles) of those calculated using , respectively.

## 6. Conclusions

Many previous studies have used alternate definitions and algorithms for computing the maximum dimension () of an ice crystal. A new method for calculating as the diameter of the smallest circle enclosing a two-dimensional () image using a linear-time algorithm is described in this study. To see the effects of different definitions, the particle size distribution (PSDs) and bulk cloud properties are derived using various definitions of . Since there is no consensus on the optimum definition of for 2D imaging probes, the uncertainties in PSDs and bulk properties due to different definitions are quantified. Derived bulk properties vary depending on the definitions of because of discrepancies in the definition of used to characterize the PSDs and that used to describe the properties of individual ice crystals. The main findings of this study are as follows:

The differences in the number distribution functions [] derived using various definitions of can differ by up to a factor of 6 for

*μ*m and mm. The large differences for*μ*m are caused by use of different definitions, as well as the strong dependence of sample volume on the particle size, whereas differences for mm are caused by the small number of particles detected.Number-weighted and mass-weighted mean diameter ( and , respectively) calculated using alternate definitions of vary from 56% to 140% and from 65% to 125% of those calculated using , respectively.

The difference in derived IWC can differ from 50% to 150% depending on the definitions of used.

The can vary from 28% to 180%, depending on the definitions of used.

The precipitation rate (mass flux) based on the above-mentioned IWC and terminal velocity calculations can differ from 20% to 250%, depending on the definitions of used.

The extinction determined using different definitions of can range from 60% to 133% of that computed using .

The effective diameter computed using different definitions of can range from 82% to 120% of that determined using .

Higher moments of PSDs have larger differences between the different definitions of than do the lower-order moments of the PSDs.

Of the six different definitions of , , , , and give smaller estimates of particle size than does , while yields a larger estimate. Using provides the closest estimate to among the six definitions considered here.

The results presented here apply only to the stratiform regions of MCSs. Further research is needed to determine how the results may vary for other kinds of clouds that may contain a different mixture of habits. In addition, the maximum dimension derived from two-dimensional images may not represent a true maximum dimension for a three-dimensional particle, unless the maximum dimension is always in a plane perpendicular to the laser beams of OAPs. Consideration of the three-dimensional value of particles would make the computation of bulk properties more complex, since the underlying *m*-*D* and *A*-*D* relations have been developed using two-dimensional projections of measured particles.

Based on the above-mentioned analysis, consistent definitions of should be used in subsequent studies deriving PSDs and bulk properties of ice clouds, because there is no simple relation converting between different definitions of (Fig. 10). Definitions that involve considerations of maximum dimensions in at least two directions (e.g., and ) are needed to get a reasonable estimate of the . The proposed in this study is an attractive choice for due to the invariant properties with respect to orientations in the imaging plane. In addition, the linear-time algorithm described in this study makes the calculation almost as fast as the calculations for other definitions. If the 3D structure of a single particle is observed in the future, the technique can be naturally extended to determine the minimum enclosing sphere, which represents the true maximum dimension of hydrometeors. Even though it is unlikely there will be a standard definition of in the near future, it is strongly suggested that the definition of used should be mentioned in subsequent papers as the uncertainties due to different definitions have been shown to be large in this study.

## Acknowledgments

The work was supported by the office of Biological and Environmental Research (BER) of the U.S. Department of Energy (DE-SC0001279, DE-SC0008500, and DE-SC0014065) as well as the National Science Foundation (NSF) (Grant AS-1213311). Data have been obtained from the ARM program archive, sponsored by the U.S. DOE Office of Science, BER, Climate and Environmental Sciences Division. The authors want to thank Michael Poellot for discussing the MC3E data quality. The discussions with Alexei Korolev about the depth of field for irregular particles and the extinction calculation directly from OAP probes improved the manuscript considerably.

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