AMTAtmospheric Measurement TechniquesAMTAtmos. Meas. Tech.1867-8548Copernicus PublicationsGöttingen, Germany10.5194/amt-10-3041-2017A statistical comparison of cirrus particle size distributions measured
using the 2-D stereo probe during the TC4, SPARTICUS, and MACPEX flight
campaigns with historical cirrus datasetsSchwartzM. Christianmcschwar@umich.eduEnvironmental Science Division, Argonne National Laboratory, 9700 Cass Avenue, Bldg. 240, 6.A.15, Lemont,
IL 60439, USAM. Christian Schwartz (mcschwar@umich.edu)23August20171083041305516February201720March201729June201730June2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://amt.copernicus.org/articles/10/3041/2017/amt-10-3041-2017.htmlThe full text article is available as a PDF file from https://amt.copernicus.org/articles/10/3041/2017/amt-10-3041-2017.pdf
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 (TC4) flight campaigns
– is used.
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.
Introduction
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 (NTs) at all times during multiple Tropical Rainfall
Measuring Mission (TRMM) field campaigns, while Field (2000) noted the same
phenomenon in midlatitude cirrus. Lawson et al. (2006) reported NTs in
midlatitude cirrus ranging from ∼ .03 to 8 cm-3 and
estimated that particles smaller than 50 µm were responsible for 99 %
of NT, 69 % of shortwave extinction, and 40 % of ice water content
(IWC). From several representative cirrus cases, Gayet et al. (2002)
reported average NTs as high as 10 cm-3 and estimated that
particles having maximum dimensions smaller than 15.8 µm resulted in
about 38 % of measured shortwave extinction, and Gayet et al. (2004, 2006) estimated from a broader set of measurements that
particles smaller than 20 µm accounted for about 35 % of observed
shortwave extinction. Garrett et al. (2003) estimated that small ice
crystals, with equivalent radii less than 30 µm, contributed in excess
of 90 % of total shortwave extinction during the NASA Cirrus Regional
Study of Tropical Anvils and Cirrus Layers – Florida Area Cirrus Experiment
(CRYSTAL-FACE).
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 µm. The basic
operating principle is thus: cloud and precipitation particles impact upon a
substrate, leaving an imprint (or leaving the particle itself) to be
replicated (in the case of the VIPS, by digital imaging) and analyzed. This
type of probe is particularly useful for imaging the smallest ice crystals
(Baumgardner et al., 2011; Brenguier et al., 2013).
Light-scattering probes also are designed for detecting small spherical and
quasi-spherical particles (a typical measurement range would be 1–50 µm;
see Baumgardner et al., 2017). These work by measuring, at various
angles, the scatter of the probe's laser due to the presence of a particle
within the probe's sample area. Assuming that detected particles are
spherical and assuming their index of refraction, Mie theory is then
inverted to estimate particle size. Two prominent examples of this type of
probe are the Forward Scattering Spectrometer Probe (FSSP; Knollenberg,
1976, 1981) and the Cloud Droplet Probe (CDP; Lance et al., 2010).
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 µm,
and the Two-Dimensional Cloud (2DC; Knollenburg, 1976) probe, whose
measurement range is 25–800 µm. OAPs designed for precipitation
particle imaging include the Precipitation Imaging Probe (PIP; Baumgardner
et al., 2001) and the High Volume Precipitation Spectrometer (HVPS; Lawson
et al., 1998), which measure particles ranging from ∼ 100 µm up to several millimeters.
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 µm and to
measure particle inter-arrival times more accurately (Lawson et al., 2010;
Korolev et al., 2013b; Brenguier et al., 2013).
Second, shattered artifacts seem mainly to corrupt particle size bins less
than about 500 µm (see also Baumgardner et al., 2011). Thus Korolev
et al. (2013b) posit that bulk quantities computed from higher-order PSD
moments – such as shortwave extinction coefficient, IWC, and radar
reflectivity – are likely to compare much better between the 2D-S and the 2DC
than is NT (see also Jackson and McFarquhar, 2014; Heymsfield et al.,
2017).
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.
Data
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 TC4, based in Costa Rica during July 2007
(TC4 Science Team, 2007). The SPEC 2D-S probe (Lawson, 2011) images ice
crystal cross sections via two orthogonal lasers that illuminate two
corresponding linear arrays of 128 photodiodes. PSDs, as well as
distributions of cross-sectional area and estimated mass, are reported every
second in 128 size bins with centers starting at 10 µm and extending
out to 1280 µm. Particles up to about 3 mm can be sized in
one dimension by recording the maximum size along the direction of flight.
During SPARTICUS the 2D-S flew aboard the SPEC Inc. Learjet, while during
MACPEX it was mounted on the NASA WB57 aircraft. During TC4 it was
mounted on both the NASA DC8 and the NASA WB57, but the WB57 data are not
used due to documented contamination of the data from shattering artifacts
off of the aircraft wing (Jensen et al., 2009).
Temperature was measured during MACPEX, TC4, and SPARTICUS using a
Rosemount total temperature probe. Bulk IWC measurements are available for
MACPEX from the Closed-path tunable diode Laser Hygrometer (CLH) probe
(Davis et al., 2007). Condensed water that enters the CLH is evaporated so
that a measurement of total water can be made. The condensed part of the
total water measured by the CLH is obtained by estimating condensed water
mass from concurrent PSDs measured by the National Center for Atmospheric
Research (NCAR) VIPS probe and then
subtracting this estimate from the measured total water mass.
Parametric fitting of PSDs
PSDs measured by the 2D-S were fit with both unimodal and bimodal parametric
gamma distributions. The unimodal distribution is
nD=N0D/D0αexp-D/D0,
where D is particle maximum dimension, D0 is the scale parameter,
α is the shape parameter, and N0 is the so-called intercept
parameter. The bimodal distribution is simply a mixture of two unimodal
distributions:
nD=N01D/D01α1exp-D/D01+N02D/D02α2exp-D/D02.
Save in a handful of instances (which will be indicated), all bulk PSD
quantities shown here are computed using these parametric fits. A
combination of unimodal and bimodal fits is used to compute NT,
dictated by the shape of the PSD as determined by a generalized chi-squared
goodness-of-fit test (Schwartz, 2014). Unimodal fits are used to compute
all other bulk quantities.
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) NTs. The difference in the number of samples
of computed NT between zero bins and one bin truncated is an order of
magnitude higher than that between one bin and two bins truncated. This is
due to frequent, extraordinarily high numbers of particles recorded in the
smallest size bin that at times cause a PSD to be flagged as bimodal by the
maximum-likelihood ratio test. As this effect lessens greatly after
truncating only one bin, and as the computed and measured NTs are
otherwise better matched using a single-bin truncation, the smallest size
bin is ignored for all PSDs (making the smallest size bin used 15–25 µm).
Also, IWC was estimated from the fit distributions (the first size bin
having been left off in the fits) using the mass–dimensional relationship
mD=0.0065D2.25 (m denotes mass, and all units are cgs)
given in Heymsfield (2003) for midlatitude cirrus. The distribution of IWC
thus computed nominally matches (not shown) IWC estimates from the both CLH
and from the 2D-S data product, which uses mass-projected area relationships
(Baker and Lawson, 2006).
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 ∼ 17 000 measured PSDs and their accompanying fits.
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, NTs computed from the remaining
bins can be underestimates. Parametric fits extrapolate the binned data all
the way to size zero, though; so it could be assumed, if the real ice
particle populations are in fact gamma-distributed, that this extrapolation
is a fair estimate of the real particles lost due to truncating the first
size bin. In truth, however, the assumption of a gamma-shaped PSD is
arbitrary, if convenient, but the gamma PSD shape is kept for its
convenience and for its ability to reproduce higher-order PSD moments.
However, in this paper – where NTs (equivalently, the zeroth moments)
from the parametric, the binned, or the normalized parametric PDSs
are computed – the computations are begun at the left edge of the second size
bin so as to compare equivalent quantities. In other words, NTs
presented for comparison here are truncated to compensate for having left
off the smallest size bin.
Normalization and parameterization
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 Deq. Before computing any moments, it is therefore necessary
first to transform all 2D-S-measured PSDs from functions of maximum
dimension D to functions of equivalent melted diameter Deq.
Each 2D-S-measured PSD nDD, whose independent variable
is ice particle maximum dimension, is transformed to a distribution
nDeqDeq, whose independent variable is equivalent
melted diameter. The transformations are performed twice: once using the
density–dimensional relationship used in D05 and once using a
mass–dimensional relationship used in D14. The first transformation allows
for application of the 2D-S data to the D05 parameterization, and the second
first transformation allows for application of the 2D-S data to the D14
parameterization.
The density–dimensional relationship ρD=aDb (ρ
denotes density, D denotes particle maximum dimension, the power law
coefficients are a=0.0056 and b=-1.1, and all units are cgs) used in
D05 stems from relationships published by Locatelli and Hobbs (1974) and
Brown and Francis (1995) for aggregate particles. Setting masses equal as in
D05 results in the independent variable transformation
Deq=aDbρw1/3D,
where ρw is the density of water.
The mass–dimensional relationship labeled “composite” (Heymsfield et al.,
2010) in D14 is used here for the second transformation:
mD=7e-3D2.2=amDbm.
(Here, m denotes mass, the power law coefficients are am=7e-3 and
bm= 2.2, and all units are cgs.) Setting masses equal results in the
independent variable transformation
Deq=6amπρw1/3Dbm/3.
The “composite” relation was only used to normalize about 54 % of the
PSDs utilized in D14; however, those datasets so normalized are broadly
similar to MACPEX, SPARTICUS, and TC4 (one in fact is TC4, where
the Cloud Imaging Probe was used as well as the 2D-S), and so the
“composite” relation is used here for comparison with D14.
Following the notation of D05 and D14 notation, transformed PSDs then have their independent
variable scaled by mass-mean diameter
Dm=∫0∞Deq4nDeqDeqdDeq∫0∞Deq3nDeqDeqdDeq
and their ordinates scaled by
N0∗=44Γ4∫0∞Deq3nDeqDeqdDeq5∫0∞Deq4nDeqDeqdDeq4,
so that
nDeqDeq=N0∗Fx=DeqDm.
In Eq. (7), F(x) is, ideally, the universal, normalized PSD (Meakin, 1992;
Westbrook et al., 2004a, b; D05; Tinel et al., 2005; D14). The quantities
N0∗ and Dm are the functions of 2D-S-measured PSD moments
that are required for application to the D05/D14 parameterizations in order
to produce parameterized, corrected 2DC PSDs and parameterized, uncorrected 2DC PSDs (see Fig. 1).
The procedure for transforming and normalizing the 2D-S-measured PSDs and
for computing N0∗ and Dm will now be explained.
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 D space to De space (see below). Then, via
numerically computed moments, Eqs. (5)–(7) are used to produce one
N0∗–Dm pair for each measured PSD and to normalize the
binned mass-equivalent spherical PSDs, which are then grouped into
normalized diameter bins of Δx= 0.10.
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 D space to Deq space, then, given that
the transformation is from maximum dimension to mass-equivalent spheres, so
also is the mass of the particles within a size bin conserved. That is,
nDeqDeqi=nDDiaDib+3ΔDiρwDeqi3ΔDeqi
for the D05 transformation and
nDeqDeqi=nDDiamDibmΔDiπ6ρwDeqi3ΔDeqi
for the D14 transformation. (The subscript i is iterated through each size
bin.)
Mass-equivalent transformations theoretically ensure that both NT and
IWC can be obtained by using the PSD in either form:
NT=∫0∞nDDdD=∫0∞nDeqDeqdDeq,
IWC=π6∫0∞aDb+3nDDdD=π6∫0∞ρwDeq3nDeqDeqdDeq,IWC=∫0∞amDbmnDDdD=π6∫0∞ρwDeq3nDeqDeqdDeq.
Whether Eq. (11a) or Eq. (11b) is used depends upon whether the D05 or the
D14 transformation is being considered. As it turns out, scaling from
D space to Deq space so that Eqs. (10) and (11) are both satisfied is not
necessarily possible. Since for the sake of estimating Dm and
N0∗ it is more important that IWCs be matched, this was done for
the D05 comparison while matching the NTs to within a factor of
approximately 0.75, plus a bias of ∼ 3.1 L-1.
The following transformation of variables must be used for computing other
bulk quantities from transformed PSDs (Bain and Englehardt, 1992):
nDD=nDeqDeqDdDeqdD.
For instance, effective radar reflectivity is computed by integrating over
particle maximum dimension intervals, using a set of particle maximum
dimension/backscatter power laws that were fit piecewise from T-matrix
computations of backscatter cross section to particle maximum dimension
(Matrosov, 2007; Matrosov et al., 2012; Hammonds et
al., 2014) as follows:
Ze=108λ4Kw2π5∑j∫DjDj+1azjDbzjnDeqDeqDdDeqdDdD.
The set of power law coefficients azj,bzj was derived
assuming an air–ice dielectric mixing model and that all particles are
prolate spheroids with aspect ratios of 0.7 (Korolev and Isaac, 2003;
Westbrook et al., 2004a, b; Hogan et al., 2012).
Several explicit expressions for computing bulk quantities based on
equivalent distributions may be found in Schwartz (2014).
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. (a) TC4, (b) MACPEX, (c) SPARTICUS, (d) all data
combined.
In D05 and D14, data taken with cloud particle and precipitation probes were
combined to give PSDs ranging from 25 µm to several millimeters. No
precipitation probe data are used here, but how does not including
precipitation probe data affect the comparison? This question will be
addressed later in this paper.
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 F∼2D-S–D05x and F∼2D-S–D14x, derived using the mass transformations of D05 and
D14, respectively. They, along with the quantities derived therefrom, serve to
parameterize the more modern 2D-S with shattered-particle removal. The
subscripts ∼ 2D-S–D05 and ∼ 2D-S–D14 are used hereinafter to represent bulk quantities derived
using F∼2D-S–D05x and F∼2D-S–D14x.
The mean, normalized PSD (D05 normalization) from all three
datasets combined, overlaid with two parameterizations from D05: the
gamma-μ parameterization (dash-dotted curve) and the modified gamma
parameterization (dashed curve). Panel (b) is a zoom-in on a portion of
panel (a).
Three parametric functions for F(x) are given in D05, two of which are repeated
here: the gamma-μ function (Fμ) and the modified gamma function
(Fα,β; Petty and Huang, 2011).
Fμx=Γ444+μ4+μΓ4+μxμexp-4+μxFα,βx=βΓ444Γα+5β4+αΓα+4β5+αxαexp-xΓα+5βΓα+4ββ
Values of μ, α, and β can be chosen to fit these
functions to a mean normalized PSD. In D05, the parametric functions
Fα,β=F-1,3 (Eq. 14) and Fμ=F3
(Eq. 13) are given to approximate the universal PSD derived from combined
2DC–2DP datasets; in D14, the parametric function Fα,β=F-0.262,1.754 is given to approximate the universal PSD
derived from shatter-corrected datasets collected mainly with combined
2DC–2DP probes.
These functions are used to parameterize transformed PSDs measured by the
2DC–2DP, given N0∗ and Dm. We therefore make the assumption
that, if we take N0∗ and Dm derived from a 2D-S-measured PSD
and then apply them to Eq. (13) or (14), we have effectively simulated the
parameterized, transformed PSD that a combined 2DC–2DP would have observed
had they been present with the 2D-S. The subscripts ∼ 2DCu and ∼ 2DCs are used
hereinafter to represent quantities that simulate 2DC–2DP data
(non-shatter-corrected and shatter-corrected, respectively) in this way.
Thus, we begin with two versions of F∼2DCux – Fμ=F3 and Fα,β=F-1,3 – and one version of F∼2DCsx: Fα,β=F-0.262,1.754. Initial observations on
comparison of F∼2D-S–D05x and F∼2D-S–D14x with F∼2DCux and F∼2DCsx will now be 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 (b)
is a zoom-in on a portion of panel (a).
Comparison with D05
Some important qualitative observations can be made from examining F∼2D-S–D05x in Fig. 4. First, in contrast to Fig. 3 of D05,
the concentrations of particles at the smallest scaled diameters of F∼2D-S–D05x are, on average, about an order of magnitude or
more lower than for the mean normalized PSD in D05. From this it is
surmised that, while the 2D-S continues to register relatively high numbers
of small ice particles, the number has decreased in the newer datasets due
to the exclusion of larger numbers of shattered ice crystals.
It can also be seen in Fig. 4 that the shoulder in the normalized PSDs in
the vicinity of x∼ 1.0 exists in the newer data as it does in
the data used in D05. It is worth noting, though, that the shoulder exists
in the one tropical dataset used here (TC4), whereas it is absent or
much less noticeable in the tropical datasets used in D05.
Fortuitously, Fα,β=F-1,3 fits the 2D-S data
better than it does the older data in D05 at the smallest normalized sizes
(cf. Fig. 2 in D05). Neither Fα,β=F-1,3
nor Fμ=F3 correctly catches the shoulder in the newer
data, though Fα,β=F-1,3 was formulated to
(better) catch a corresponding shoulder in the older data.
Total number concentration computed using the parameterized
universal PSDs from D05 along with true values of N0∗ and
Dm (from the 2D-S data) scattered vs. total number concentration
computed directly from untransformed 2D-S data.
Next, a comparison of PSD quantities computed directly from the 2D-S with
corresponding ∼ 2DC-derived quantities (computed using
N0∗ and Dm derived directly from the binned 2D-S data and
applied to Fα,β=F-1,3 and Fμ=F3) is made. The extinction coefficient, IWC, and 94 GHz radar
reflectivity compare well between the 2D-S and both versions of
∼ 2DCu (not shown). As for NT, it is the least certain computation (see
Fig. 7), but Fμ=F3 is entirely wrong in attempting to reproduce this
quantity, so this shape is not used hereinafter, and F∼2DCux=F-1,3x is the shape used to
simulate the uncorrected 2DC–2DP.
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
∼ 2DCu-derived quantities. Effective radius is as defined in D05. Mean relative
error for both extinction coefficient and IWC is about -0.1 %. The mean
relative error in NT (NT computed directly from truncated, binned
PSDs is used both here and in Fig. 9) is rather large at ∼ 50 %;
the mean relative error in Ze, at ∼ 22 %,
is larger than that shown in Fig. 5 of D05 (less than 5 % there) but, at
about 2 dB, is within the error of most radars. This may well be due to the
overestimation of Fx by F∼2DCux
between normalized sizes of about 1.2 and 2 (see Fig. 4b). Both here and in
D05, F∼2DCux falls off much more rapidly than
F∼2D-S–D05x above a normalized diameter of 2.
However, it is deduced from Figs. 2 and 5 in D05 that this roll-off is not
responsible for the large mean relative error in Z shown in Fig. 8.
Mean relative error and standard deviation of the relative error
between total number concentration (divided by 10), effective radius, IWC,
and Z as computed directly from the 2D-S and as computed from the
modified-gamma universal PSD shape and the true N0∗ and Dm
computed from the 2D-S data. Standard error of the mean and standard
deviation are shown with red error bars.
As in Fig. 8 but using the shatter-corrected 2DC
parameterization.
The mean relative error in effective radius shown in Fig. 8 is approximately
-7 %, whereas it is apparently nil in Fig. 5 of D05. Effective radius is
defined in D05 as the ratio of the third to the second moments of the
spherical-equivalent PSDs and is therefore a weighted mean of the PSD. The
negative sign on the relative error indicates that, on average, F∼2DCux is underestimating the effective radius of the
PSDs measured by the 2D-S, whereas for the older datasets it hits the
effective radius spot on (in the average). Therefore, there is a significant
difference between the 2D-S datasets and the older 2DC–2DP datasets in the
ratio of large particles to small particles, even when precipitation probe
data are not combined with the 2D-S.
Comparison with D14
From Fig. 5, concentrations at the smallest scaled diameters of F∼2D-S–D14x are nominally consistent with
those shown in Fig. 6 of D14. In accordance with the surmise made in the comparison with D05
above, it would seem that shattered-particle removal from the 2DC improves
comparison between the 2D-S and the 2DC–2DP at the smallest particle sizes.
Here, F∼2DCsx=F-0.262,1.754x. The shoulder in the normalized PSDs in the vicinity of x∼ 1.0 is again found, though the shoulder is not captured by
F∼2DCsx (see Fig. 6). The normalized 2D-S at the
smallest normalized sizes is also underestimated by F∼2DCsx. Comparison of NT computed using F∼2DCsx with that derived from 2D-S is quite similar to that of F∼2DCux (not shown).
Data from TC4 alone. The mean normalized PSD from the 2D-S
is overlaid with the mean normalized PSD obtained from combining the 2D-S
with the PIP and the modified gamma parameterization from D05 (dashed
curve). Panel (b) is a zoom-in on a portion of panel (a).
As shown in Fig. 9, the mean relative error between NT and effective
radius derived from the 2D-S and from ∼ 2DCs is again about 50 %, while the mean
relative error in effective radius remains about -7.5 %. The mean relative
error in reflectivity has decreased to about 14 %.
Impact of not using precipitation probe data
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 TC4
dataset. This campaign of the three was chosen due to its tending to occur
at warmer temperatures, in a more convective environment, and at lower
relative humidities: therefore, if large particles are going to matter, they
should matter for TC4. Figure 10 shows, similar to Figs. 4 and 6,
F∼2D-S–D05x for the 2D-S alone, F∼2D-S/PIP–D05x for the 2D-S combined with the PIP, and
F∼2DCux.
In the combined data, F∼2D-S/PIP–D05x does not dig as
low between zero and unity as for the 2D-S alone, but it does show similar
numbers of particles at the very smallest normalized sizes, and the shoulder
is at the same location. Beginning at about x= 1.2, the 2D-S/PIP
normalized distribution is higher than the 2D-S-alone normalized
distribution, and it continues out to about x= 10, whereas the 2D-S-alone
distribution ends shy of x= 5. In either case, F∼2DCux misses what is greater than about x= 2. This roll-off, along
with the fact that F∼2D-S/PIP–D05x appears to be more
similar to F∼2D-S–D05x than it does to
F∼2DCux, indicates that a parameterization ofFx based off the 2D-S alone is comparable to the 2DC/2DP-based
F∼2DCux parameterization.
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 N0∗
and Dm computed from the 2D-S alone and from the 2D-S combined with the
PIP. (a)NT, (b) extinction coefficient, (c) IWC, (d) 94 GHz effective
radar reflectivity.
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)
N0∗ and Dm computed from each of the 2D-S PSDs alone and
from the combined PSDs from TC4 were used, along with F∼2DCux, to compute NT, extinction coefficient, IWC, and
94 GHz effective radar reflectivity. This amounts to two different
∼ 2DCu simulations: one including the PIP and one not. The results are shown in
Fig. 12. The distributions are very similar, with the exception of the
reflectivity distributions, whose means are separated by less than 1 dBZ. It
is concluded that the cloud filtering technique has resulted in PSDs that
are satisfactorily described by the 2D-S alone, at least in the case of this
comparison.
Marginal PDFs of quantities computed directly from 2D-S data, as
well as computed using the parameterized 2D-S and the parameterized,
uncorrected 2DC. (a) Total number concentration, (b) shortwave extinction
coefficient, (c) ice water content, (d) radar reflectivity.
Marginal PDFs of quantities computed directly from 2D-S data, as
well as computed using the parameterized 2D-S and the parameterized,
corrected 2DC. (a) Total number concentration, (b) shortwave extinction.
Final results and discussion
In D05, complete parameterization of a 2DC–2DP-measured PSD is achieved by
using the universal shape Fα,βx
along with N0∗ parameterized by radar reflectivity and Dm
parameterized by temperature. For comparison with the shattered-corrected
D14 study, a temperature-based parameterization of “composite”-derivedDm
is also computed from the 2D-S data, and “composite”-derived N0∗ is also parameterized by radar reflectivity. A similar parameterization
scheme (also based on radar reflectivity and temperature) for the 2D-S
(based on Field et al., 2005) is outlined in Schwartz (2014) and is used
here to compute a fully parameterized version of 2D-S-measured PSDs so as to
make a fair comparison of them with fully parameterized 2DC–2DP-measured
PSDs.
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 Dm. The PDFs of extinction coefficient and IWC for
the two parameterized instruments match one another quite well (the
differences in their medians are not statistically significant). However,
for NT, the x2DCu PDF is shifted to higher concentrations than the PDF for
x2D-S. The difference in their medians is statistically significant at the 95 %
level according to a Mann–Whitney U test. It is therefore concluded that the
older D05 parameterization based on the 2DC–2DP datasets predicts a
statistically significantly higher number of total ice crystals than does the
parameterized 2D-S (by a factor of about 1.3, or a little over 1 dB) and
that, more generally, the 2DC measures a larger ratio of small ice crystals
to large ice crystals than does the 2D-S, as shown in the effective radius
comparison in Fig. 8.
Figure 14 shows PDFs of NT and extinction coefficient computed using
the fully parameterized 2D-S (red, labeled “x2D-S”), using the fully parameterized
(corrected) 2DC–2DP (blue, labeled “x2DCs”), and directly from the 2D-S data
(black). The PDFs of extinction match quite well, but their medians are
significantly different according to the U test. The medians of NT are
also significantly different, but the mean of the parameterized, corrected
2DC is lower than that of the parameterized 2D-S. A posteriori shatter
correction has made 2DC measurements more like 2D-S measurements in the bulk
quantity of total particle concentration; however, a statistically
significant difference between the 2D-S and the corrected 2DC remains. This
result is entirely expected in light of the previous results outlined in the
Introduction.
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 NT – instrument improvements may have produced
only marginal improvements.
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 µm, whereas the
2D-S data used here are truncated to begin at 15 µm. This means that the
2D-S data had the potential of measuring greater numbers of small particles
than did the 2DC, and yet the differences in small particles between D05 and
the current study were still realized.
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 TC4 data
may be accessed from the NASA Earth Science Project Office (ESPO) data
archive, also noted in the references.
The author declares that he has no conflict of interest.
Acknowledgements
The author gratefully acknowledges the SPARTICUS, MACPEX, and TC4
science teams for the collection of data used in this study. TC4 and
MACPEX data were obtained from the NASA ESPO archive, which may be accessed
online at https://espoarchive.nasa.gov/archive/browse/. The SPARTICUS data
were obtained from the Department of Energy (DOE) ARM archive and may be accessed
online at http://www.archive.arm.gov/discovery/#v/results/s/fiop::aaf2009Sparticus.
In particular, the author acknowledges Paul Lawson and SPEC Inc. for
all 2D-S data collected in the field, Andrew Heymsfield for the PIP
data used from TC4, and Linnea Avallone for CLH data used from
MACPEX. Thanks are furthermore given to Gerald G. Mace, Paul Lawson,
and Andrew Heymsfield for helpful discussions that led to significant
improvements in the manuscript. This work was supported by the US
DOE Atmospheric System Research (ASR) – an Office of
Science, Office of Biological and Environmental Research (BER) program –
under contract DE-AC02-06CH11357 awarded to Argonne National Laboratory.
This work was also supported by the National Science Foundation (NSF) grant
AGS-1445831. Grateful acknowledgement is also given to the computing
resources provided on Blues, a high-performance computing cluster operated
by the Laboratory Computing Resource Center (LCRC) at the Argonne National
Laboratory.
Edited by: Alexander Kokhanovsky
Reviewed by: Darrel Baumgardner, Jeffrey Reid, and two anonymous referees
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