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 μm, with the exact size range of each probe depending on the magnification, resolution, and distance between probe arms. High-resolution (2.3 μm) 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 (
List of the definitions of particle dimension used in this article.

Even if maximum dimension is used to represent PSDs, there are different methods that have been used to calculate
There are several different ways
The methods for calculating the
2. Definitions of Dmax
In this section, different methods of determining

Schematic of definition of
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0177.1
In the past, at least five different definitions of
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(n4) 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
3. Dataset
To test the newly implemented Gärtner (1999) algorithm for calculating
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.

Flight track of UND Citation on 20 May 2011. The color indicates the time during the flight.
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0177.1
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.

Representative particles imaged by the 2D-C during four flight legs between 1330 and 1530 CDT 20 May 2011 shown as a function of average temperature of leg on which they were measured.
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0177.1
The CIP, 2D-C, and HVPS were installed on the UND Citation, and nominally sampled particles between 25 μm 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

Overlap of average PSDs measured by 2D-C and HVPS for all time with T < 0°C for MCS sampled on 20 May 2011.
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0177.1
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

The distribution of interarrival time for (a) 2D-C and (b) HVPS for time when constant-altitude flight legs were flown by UND Citation with T < 0°C.
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0177.1




List of m-D relation (













The sample volume calculated using center-in, entire-in, and HP78 extension for 2D-C, CIP, and HVPS, installed on UND Citation during MC3E.
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0177.1






















Besides the bulk properties addressed here, there are also differences in the radar reflectivity derived from PSDs using alternate definitions of
For calculating quantities in Eqs. (5)–(12), the measured particles are first sorted into bins of varying width with
4. Effect of 
definitions on PSDs

The composite PSDs computed using six different definitions of

Term
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0177.1
The trends in how the different definitions of
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
When comparing the behavior of PSDs for different temperatures in Fig. 7, it is apparent that the differences in
The impact of the different definitions of

Box-and-whisker plot showing ratio of (a)
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0177.1
Summary of the median ratio of bulk properties derived using various of definitions of

Two factors contribute to the differences between
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
a. Total number concentration
The

Box-and-whisker plot showing ratio of bulk property: (a) total number concentration, (b) IWC, (c) mass-weighted terminal velocity, (d) precipitation rate, (e) extinction, and (f) effective diameter computed using definition of
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0177.1
b. Ice water content
The IWCs calculated using different definitions of
For the implementation of an m-

Relations between
Citation: Journal of Atmospheric and Oceanic Technology 33, 5; 10.1175/JTECH-D-15-0177.1
c. Mass-weighted terminal velocity
Figure 9c shows that
d. Extinction
Figure 9e shows the variation in extinction determined using different definitions of
e. Effective diameter
Figure 9f shows the effective diameter calculated using the different definitions of
6. Conclusions
Many previous studies have used alternate definitions and algorithms for computing the maximum dimension (
- The differences in the number distribution functions [
] derived using various definitions of can differ by up to a factor of 6 for μm and mm. The large differences for μm 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
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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