This paper addresses two straightforward questions. First, how
similar are the statistics of cirrus particle size distribution (PSD)
datasets collected using the Two-Dimensional Stereo (2D-S) probe to cirrus PSD datasets
collected using older Particle Measuring Systems (PMS) 2-D Cloud (2DC) and 2-D
Precipitation (2DP) probes? Second, how similar are the datasets when
shatter-correcting post-processing is applied to the 2DC datasets? To answer
these questions, a database of measured and parameterized cirrus PSDs –
constructed from measurements taken during the Small Particles in Cirrus
(SPARTICUS); Mid-latitude Airborne Cirrus Properties Experiment (MACPEX); and
Tropical Composition, Cloud, and Climate Coupling (TC

Bulk cloud quantities are computed from the 2D-S database in three ways: first, directly from the 2D-S data; second, by applying the 2D-S data to ice PSD parameterizations developed using sets of cirrus measurements collected using the older PMS probes; and third, by applying the 2D-S data to a similar parameterization developed using the 2D-S data themselves. This is done so that measurements of the same cloud volumes by parameterized versions of the 2DC and 2D-S can be compared with one another. It is thereby seen – given the same cloud field and given the same assumptions concerning ice crystal cross-sectional area, density, and radar cross section – that the parameterized 2D-S and the parameterized 2DC predict similar distributions of inferred shortwave extinction coefficient, ice water content, and 94 GHz radar reflectivity. However, the parameterization of the 2DC based on uncorrected data predicts a statistically significantly higher number of total ice crystals and a larger ratio of small ice crystals to large ice crystals than does the parameterized 2D-S. The 2DC parameterization based on shatter-corrected data also predicts statistically different numbers of ice crystals than does the parameterized 2D-S, but the comparison between the two is nevertheless more favorable. It is concluded that the older datasets continue to be useful for scientific purposes, with certain caveats, and that continuing field investigations of cirrus with more modern probes is desirable.

For decades, in situ ice cloud particle measurements have often indicated
ubiquitous high concentrations of the smallest ice particles (Korolev et
al., 2013a; Korolev and Field, 2015). If the smallest ice particles are
indeed always present in such large numbers, then their effects on cloud
microphysical and radiative properties are pronounced. For instance,
Heymsfield et al. (2002) reported small particles dominating total particle
concentrations (

While it is quite possible for relatively high numbers of small ice crystals to occur naturally (see, e.g., Zhao et al., 2011; Heymsfield et al., 2017), it is also possible for small-ice-particle concentrations to be significantly inflated by several measurement artifacts. The various particle size distribution (PSD) probes (also known as single-particle detectors) in use employ a handful of different measurement techniques to detect and size particles across a variety of particle size ranges. The units of a PSD are number of particles per unit volume per unit size. Thus, after a PSD probe counts the particles that pass through its sample area, each particle is assigned a size as well as an estimate of the sample volume from which it was drawn (Brenguier et al., 2013). Uncertainty in any of these PSD components results in uncertain PSD estimates.

Leaving aside technologies still under development and test, such as the
Holographic Detector of Clouds (HOLODEC; Fugal and Shaw, 2009), PSD probes
fall into three basic categories: impactor probes, light-scattering probes,
and imaging probes. (More thorough discussions on this topic, along with
comprehensive bibliographies, may be found in Brenguier et al. (2013) and in
Baumgardner et al. (2017).) The earliest cloud and precipitation particle
probes were of the impactor type (Brenguier et al., 2013). Modern examples
include the Video Ice Particle Sampler (VIPS; Heymsfield and McFarquhar,
1996), designed to detect particles in the range 5–200

Light-scattering probes also are designed for detecting small spherical and
quasi-spherical particles (a typical measurement range would be 1–50

Imaging probes, also known as optical array probes (OAPs), use arrays of
photodetectors to make two-dimensional images of particles that pass through
their sample areas. Unlike the light-scattering probes, OAPs make no
assumptions regarding particle shape or composition (Baumgardner et al.,
2017), and they have broader measurement ranges aimed both at cloud and
precipitation particles. Two prominent examples are the Two-Dimensional
Stereo (2D-S; Lawson et al., 2006) probe, whose measurement range is 10–1280

Because an estimate of the sample volume from which a particle is drawn is a function of the particle's size and assumes that the particle is spherical (Brenguier et al., 2013), all PSD probes suffer from sample volume uncertainty. Estimated sample volumes from OAPs perforce suffer from the problem of sizing aspherical particles from 2-D images (see Figs. 5–40 of Brenguier et al., 2013). Nonetheless, impactor and light-scattering probes both suffer from much smaller sample volumes than do OAPs (Brenguier et al., 2013; Baumgardner et al., 2017; Heymsfield et al., 2017). Scattering probes, for example, need up to several times the sampling distance in cloud as OAPs to produce a statistically significant PSD estimate (see Figs. 3–5 of Brenguier et al., 2013).

The obvious difficulty in sizing small ice crystals with light-scattering probes is the application of Mie theory to nonspherical ice crystals. Probes such as the FSSP and CDP are therefore prone to undersizing ice crystals (Baumgardner et al., 2011, 2017; Brenguier et al., 2013).

Imaging particles using an OAP requires no assumptions regarding particle shape or composition, but sizing algorithms based on two-dimensional images are highly sensitive to particle orientation (Brenguier et al., 2013). Other sizing uncertainties stem from imperfect thresholds for significant occultation of photodiodes, the lack of an effective algorithm for bringing out-of-focus ice particles into focus, and the use of statistical reconstructions of partially imaged ice crystals that graze a probe's sample area (Brenguier et al., 2013; Baumgardner et al., 2017).

Ideally, PSDs estimated using different probes would be stitched together in order to provide a complete picture of the ice particle population, from micron-sized particles through snowflakes (Brenguier et al., 2013). However, while data from VIPS, fast FSSP, and Small Ice Detector-3 (SID-3; Ulanowski et al., 2014) probes are available to complement the OAP data used in this study, none of them are used on account of sizing uncertainties stemming from their small sample volumes and from spherical particle assumptions. The two publications wherewith comparison is made in this paper also restricted their datasets to OAPs.

The substantial remaining source of small particle counting and sizing dealt with in this study is particle shattering. Shattering of ice particles on probe tips and inlets and on aircraft wings has rendered many historical cirrus datasets suspect (Vidaurre and Hallet, 2009; Korolev et al., 2011; Baumgardner et al., 2017) due to such shattering artificially inflating measurements of small-ice-particle concentrations (see, e.g., McFarquhar et al., 2007; Jensen et al., 2009; Zhao et al., 2011). Measured ice PSDs are used to formulate parameterizations of cloud processes in climate and weather models, so the question of the impact of crystal shattering on the historical record of ice PSD measurements is one of significance (Korolov and Field, 2015).

Post-processing of optical probe data based on measured particle inter-arrival times (Cooper, 1978; Field et al., 2003, 2006; Lawson, 2011; Jackson et al., 2014; Korolev and Field, 2015) has become a tool for ameliorating contamination from shattered artifacts. Shattered-particle removal is based on modeling particle inter-arrival times by a Poisson process, assuming that each inter-arrival time is independent of all other inter-arrival times. Jackson and McFarquhar (2014) posit that particle clustering (Hobbs and Rangno, 1985; Kostinski and Shaw, 2001; Pinsky and Khain, 2003; Khain et al., 2007), which would violate this basic assumption, is not likely a matter of significant concern as cirrus particles are naturally spread further apart than are liquid droplets and sediment over a continuum of size-dependent speeds.

In addition, a posteriori shattered-particle removal should be augmented with design measures such as specialized probe arms and tips (Vidaurre and Hallet, 2009; Korolov et al., 2011, 2013a; Korolev and Field, 2015). Probes must also be placed away from leading wing edges (Vidauure and Hallet, 2009; Jensen et al., 2009), as many small particles generated by shattering on aircraft parts are likely not be filtered out by shatter-recognition algorithms.

The ideal way to study the impact of both shattered-particle removal and improved probe design is to fly two versions of a probe – one with modified design and one without – side by side and then to compare results from both versions of the probe both with and without shattered-particle removal. Results from several flight legs made during three field campaigns where this was done are described in three recent papers: Korolev et al. (2013b), Jackson and McFarquhar (2014), and Jackson et al. (2014). Probes built for several particle size ranges were examined, but those of interest here are the 2D-S and the older 2DC. Three particular results distilled from those papers are useful here.

First, in agreement with Lawson (2011), a posteriori shattered-particle
removal is more effective at reducing counts of apparent shattering
fragments for the 2D-S than are modified probe tips. The opposite is true
for the 2DC. This is attributed to the 2D-S's larger sample volume; to its
improvements in resolution and electronic time response over the 2DC; and to
its 256 photodiode elements (Jensen et al., 2009; Lawson, 2011; Brenguier et
al., 2013), which allow it to size particles smaller than 100

Second, shattered artifacts seem mainly to corrupt particle size bins less
than about 500

Third, the efficacy of shattered-particle removal from the 2DC is questionable: the post-processing is prone to accepting shattered particles and to rejecting real particles (Korolev and Field, 2015). The parameters of the underlying Poisson model and its ability to correctly identify shattered fragments depend on the physics of the cloud being sampled (Vidaurre and Hallett, 2009; Korolev et al., 2011), and the older 2DC experiences more issues with instrument depth of field, unfocused images, and image digitization than do newer OAPs, further compounding uncertainty in the shattered-particle removal (Korolev et al., 2013b; Korolev and Field, 2015).

In the context of relatively small studies such as these, Korolev et al. (2013b) pose two questions: (i) to what extent can the historical data be used for microphysical characterization of ice clouds, and (ii) can the historical data be reanalyzed to filter out the data affected by shattering? One difficulty in addressing these questions is the scarcity of data from side-by-side instrument comparisons. Another is that, especially for the 2DC, “correcting [data] a posteriori is not a satisfactory solution” (Vidaurre and Hallet, 2009). However, shattered-particle removal is the main (if not the only) correction method available when revisiting historical datasets.

In order to address the first question of Korolev et al. (2013b), bulk cloud properties derived from shatter-corrected 2D-S data are used to answer two questions: (1) how similar are the statistics of cirrus PSD datasets collected using the 2D-S probe to cirrus PSD datasets collected using older 2DC and 2DP (2-D Precipitation) probes? (2) How similar are the datasets when shatter-correcting post-processing is applied to the 2DC datasets? In proceeding, two points are critical to recall. First, the 2D-S is reasonably expected to give results superior to the 2DC after shattered-particle removal. Second, lingering uncertainty notwithstanding, results presented elsewhere from the shatter-corrected 2D-S reveal behaviors in ice microphysics within different regions of cloud that are expected both from physical reasoning and from modeling studies and that were not always discernible before from in situ datasets (Lawson, 2011; Schwartz et al., 2014).

Flowchart illustrating the method of comparison between the parameterized shatter-corrected 2DC–2DP dataset, uncorrected 2DC–2DP dataset, and shatter-corrected 2D-S dataset.

To this end, a substantial climatology of shatter-corrected, 2D-S-measured cirrus PSDs is indirectly compared with two large collections of older datasets, collected from the early 1990s through the mid-2000s mainly using Particle Measurement Systems 2DC and 2DP (Baumgardner, 1989) as well as Droplet Measurement Technologies Cloud Imaging Probe (CIP) and PIP instruments (Heymsfield et al., 2009) and, in one instance, the 2D-S. The older datasets are presented and parameterized in Delanoë et al. (2005; hereinafter D05) and in Delanoë et al. (2014; hereinafter D14). The data used in D05 were not subject to shattered-particle removal, whereas the data in D14 were a posteriori.

The comparison strategy, in short is as follows. The D05 and D14 parameterizations consist of normalized, “universal” cirrus PSDs to which functions of PSD moments are applied as inputs. The results of so doing are sets of parameterized 2DC PSDs – both shatter-corrected and uncorrected. To make the comparison, the same moments from 2D-S-measured PSDs are applied to the D05 and D14 parameterizations in order to simulate what the shatter- and non-shatter-corrected 2DCs would have measured had they flown with the 2D-S. Then, a universal PSD derived from the 2D-S itself is computed in order to make a fair comparison. The moments from the 2D-S-measured PSDs are applied to the 2D-S universal PSD, and it is then seen whether the older datasets differ statistically from the newer in their derived cirrus bulk properties. This procedure is illustrated in Fig. 1.

Section 2 contains a description of the data used herein. Section 3 discusses the fitting of PSDs with gamma distributions for computational use, Sect. 4 discusses the normalization and parameterization schemes used by D05 and D14, and Sect. 5 discusses the effects of not having included precipitation probe data with the 2D-S data. Section 6 demonstrates the final results of the comparison and concludes with a discussion.

The 2D-S data were collected during the Mid-Latitude Airborne Cirrus
Experiment (MACPEX), based in Houston, TX, during February and March 2011
(MACPEX Science Team, 2011); the Small Particles in Cirrus (SPARTICUS)
campaign, based in Oklahoma during January through June 2010
(SPARTICUS Science Team, 2010); and TC

Temperature was measured during MACPEX, TC

PSDs measured by the 2D-S were fit with both unimodal and bimodal parametric
gamma distributions. The unimodal distribution is

Unimodal fits were performed via the method of moments (in a manner similar to Heymsfield et al., 2002). Both the method of moments and an expectation maximization algorithm (Moon, 1996; Schwartz, 2014) were used for the bimodal fits – the more accurate of those two fits (as determined by whether fit provided the smaller binned Anderson–Darling test statistic; Demortier, 1995) being kept.

Comparisons of computed and measured total number concentration for 15 s PSD averages and for truncation of none through the first two PSD size bins.

Measured PSDs are both truncated and time-averaged in order to mitigate counting uncertainties. It is here assumed that temporal averaging sufficiently reduces Poisson counting noise so that it may be ignored (see, e.g., Gayet et al., 2002). Given already-cited concerns regarding uncertainty in shattered-particle removal, the smallest size bins are not automatically assumed here to be reliable. Other competing uncertainties further complicate particle counts within the first few size bins, e.g., decreased detection efficiency within the first size bin (Baumgardner et al., 2017), the possible underestimation of counts of real particles by a factor of 5–10 (Gurganus and Lawson, 2016), and mis-sizing of larger particles into smaller size bins due to image breakup at the edge of the instrument's depth of field (Korolev et al., 2013b; Korolev and Field, 2015; Baumgardner et al., 2017).

In order to determine how many of the smallest size bins to truncate and for
how many seconds to average in order to make the counting assumption valid,
two simple exercises were performed using the MACPEX dataset. In the first
exercise, 15 s temporal averages were performed along with
truncating zero through two of the smallest size bins while only the
unimodal fits (chosen according to a maximum-likelihood ratio test; Wilks,
2006) were kept. This exercise was performed first so as to prevent the
most spurious size bins interfering with the smoothing out of Poisson
counting noise. Figure 2 shows comparisons of distributions of measured and
computed (from the fits)

Also, IWC was estimated from the fit distributions (the first size bin
having been left off in the fits) using the mass–dimensional relationship

For the second exercise, temporal averages from 1 to 20 s were
performed, truncating the first size bin and again keeping only the unimodal
fits. The balance to strike in picking a temporal average length is to
smooth out Poisson counting uncertainties acceptably without losing physical
information to an overlong average. Qualitatively, the statistics of the fit
parameters begin to steady at around 15 s (not shown), so a
15 s temporal average was chosen. Using the data filters, temporal
average, and bin truncation thus far described results in

It must be noted that the first 2D-S size bin contains at least some real
particles, though the aforementioned uncertainties make it impossible (at
present) to know how many. Therefore,

In this section, the functions of 2D-S-measured PSD moments that are applied
to the D05 and D14 parameterizations (see Fig. 1) are explained. However, the
D05 and D14 parameterizations make use of PSDs in terms of equivalent melted
diameter

Each 2D-S-measured PSD

The density–dimensional relationship

The mass–dimensional relationship labeled “composite” (Heymsfield et al.,
2010) in D14 is used here for the second transformation:

Following the notation of D05 and D14 notation, transformed PSDs then have their independent
variable scaled by mass-mean diameter

In Eq. (7),

Starting with binned PSDs, the normalization procedure is wended as
described in Sect. 4.1 of D05. First, the 2D-S bin centers and bin widths
are transformed once using Eq. (3) for the comparison with D05 and once
again using Eq. (4) for the comparison with D14. Next, each binned PSD is
transformed by scaling from

The scale factor for transforming binned PSDs is derived using this simple
consideration: if the number of particles within a size bin is conserved
upon the bin's transformation from

Mass-equivalent transformations theoretically ensure that both

The following transformation of variables must be used for computing other
bulk quantities from transformed PSDs (Bain and Englehardt, 1992):

Histograms of normalized PSDs from each flight campaign, overlaid
with their mean, normalized PSDs (D05 normalization). The color map is
truncated at 75 % of the highest number of samples in a bin so as to
increase contrast.

In D05 and D14, data taken with cloud particle and precipitation probes were
combined to give PSDs ranging from 25

Two-dimensional histograms of the normalized PSDs are shown in Fig. 3 for
the D05 transformation and in Fig. 5 for the D14 transformation, overlaid
with their mean normalized PSDs (cf. Figs. 1 and 2 in D05 and Fig. 3 in
D14). For both transformations, the mean normalized PSDs for the three
datasets combined are repeated in Figs. 4 and 6 as solid curves (cf. Fig. 3
of D05 and Fig. 6 of D14). These serve as the empirical universal,
normalized PSDs

The mean, normalized PSD (D05 normalization) from all three
datasets combined, overlaid with two parameterizations from D05: the
gamma-

Three parametric functions for

These functions are used to parameterize transformed PSDs measured by the
2DC–2DP, given

Same as Fig. 3 but using D14 normalization.

The mean normalized PSD (D14 normalization) from all three
datasets combined, overlaid with the parameterizations from D14.
Panel

Some important qualitative observations can be made from examining

It can also be seen in Fig. 4 that the shoulder in the normalized PSDs in
the vicinity of

Fortuitously,

Total number concentration computed using the parameterized
universal PSDs from D05 along with true values of

Next, a comparison of PSD quantities computed directly from the 2D-S with
corresponding

Figure 8 shows the mean relative error and the standard deviation of the
relative error (cf. Fig. 5 of D05) between 2D-S-derived and corresponding

Mean relative error and standard deviation of the relative error
between total number concentration (divided by 10), effective radius, IWC,
and

As in Fig. 8 but using the shatter-corrected 2DC parameterization.

The mean relative error in effective radius shown in Fig. 8 is approximately

From Fig. 5, concentrations at the smallest scaled diameters of

Here,

Data from TC

As shown in Fig. 9, the mean relative error between

To more formally investigate the impact of not using a precipitation probe,
data from the PIP were combined with data from the 2D-S using the TC

In the combined data,

Two-dimensional histogram of 94 GHz effective radar reflectivity computed, using the Hammonds–Matrosov approach, from the 2D-S alone versus that computed from the 2D-S combined with the PIP.

Distributions of quantities computed using the parametric
modified gamma distribution along with the true values of

In support of this assertion, Fig. 11 shows the penalty in radar reflectivity, computed directly from data using the approach described earlier, incurred by using only the 2D-S instead of the 2D-S-PIP. The penalty is in the neighborhood of 1 dB.

The true (in the sense that they are derived directly from measurements)

Marginal PDFs of quantities computed directly from 2D-S data, as
well as computed using the parameterized 2D-S and the parameterized,
uncorrected 2DC.

Marginal PDFs of quantities computed directly from 2D-S data, as
well as computed using the parameterized 2D-S and the parameterized,
corrected 2DC.

In D05, complete parameterization of a 2DC–2DP-measured PSD is achieved by
using the universal shape

Figure 13 shows the results of computing PSD-based quantities using the
fully parameterized 2D-S (red, labeled “x2D-S”), using the fully parameterized
(uncorrected) 2DC–2DP (blue, labeled “x2DCu”), and directly from the 2D-S data
(black). Probability density functions (PDFs) of 94 GHz effective radar
reflectivity match because they are forced to by the two instrument
parameterizations. Otherwise, biases exist between the two sets of
computations based on simulated instruments and computations based on the
actual 2D-S (black curve). This bias is due mainly to the temperature
parameterization of

Figure 14 shows PDFs of

In this paper, an indirect comparison to older 2DC-based datasets by means of parameterizations given in D05 and in D14 has been made. The main discussion points and some sources of uncertainty are now enumerated.

It is determined that the 2D-S cirrus cloud datasets used here are significantly different from historical datasets in numbers of small ice crystals measured. With a posteriori shattered-particle removal applied to older 2DC data, the total numbers of ice crystals measured by the 2D-S and the 2DC become more similar, but NT measured by the 2DC remains statistically different from that measured by the 2D-S.

Given the modest differences found here between bulk cirrus properties
derived from PSDs, we conclude that historical datasets continue to be
useful. It would seem that for the measurement of bulk cirrus
properties – excepting

It is surmised that – since the efficacy of a posteriori shatter correction on the 2DC is questionable; since the 2D-S is superior in response time, resolution, and sample volume to the 2DC; and since steps were taken to mitigate ice particle shattering on the 2D-S data – the newer datasets are more accurate. Therefore, continuing large-scale field investigations of cirrus clouds using newer particle probes and data processing techniques is recommended. Where possible, investigation of the possibility of statistical comparison and correction of historical cirrus ice particle datasets using newer datasets by flying 2DC probes alongside 2D-S and other more advanced probes is strongly encouraged.

There are some sources of uncertainty.

There exists a large amount of uncertainty in mass–dimensional and density–dimensional relationships for ice crystals, such as those used in D05, in D14, and in this paper. In making a comparison, the best that could be done was to use the same relations in this paper as in D05 and D14. This, of course – depending on which part of the comparison is considered – assumes that the same overall mix of particles habits was encountered between D05 and this study and between D14 and this study.

The data for both D05 and D14 are stated to begin at 25

Finally, it is important to note that this study does not specifically consider PSD shape. (For a more detailed discussion on cirrus PSD shape and on the efficacy of the gamma distribution, please refer to Schwartz, 2014.) This is a critical component of the answers to the original two questions of Korolov et al. (2013b). Mitchell et al. (2011) demonstrated that, for a given effective diameter and IWC, the optical properties of a PSD are sensitive to its shape. Therefore, PSD bimodality and concentrations of small ice crystals are critical to realistically parameterizing cirrus PSDs, to modeling their radiative properties and sedimentation velocities, and to mathematical forward models designed to infer cirrus PSDs from remote-sensing observations (Lawson et al., 2010; Mitchell et al., 2011; Lawson, 2011). In order to improve knowledge on PSD shape, as well as to develop statistical algorithms for correcting historical PSD datasets so that PSD shapes are corrected along with computations of bulk properties, it will be necessary to make use of instruments that can provide reliable measurements of small ice crystals beneath the size floors of both the 2DC and the 2D-S. Recent studies such as Gerber and DeMott (2014) have provided aspherical correction factors for particle volumes and effective diameters measured by the FSSP. However, the author expects that this problem will ultimately be resolved by the continued technological development of new probes such as the HOLODEC.

All SPARTICUS data may be accessed via the Atmospheric Radiation Measurement
(ARM) data archive as noted in the references. All MACPEX and TC

The author declares that he has no conflict of interest.

The author gratefully acknowledges the SPARTICUS, MACPEX, and TC