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- About
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AMT | Articles | Volume 12, issue 4

Atmos. Meas. Tech., 12, 2261-2285, 2019

https://doi.org/10.5194/amt-12-2261-2019

© Author(s) 2019. This work is distributed under

the Creative Commons Attribution 4.0 License.

https://doi.org/10.5194/amt-12-2261-2019

© Author(s) 2019. This work is distributed under

the Creative Commons Attribution 4.0 License.

Special issue: CALIPSO version 4 algorithms and data products

**Research article**
12 Apr 2019

**Research article** | 12 Apr 2019

Application of high-dimensional fuzzy *k*-means cluster analysis

^{1}Science Systems and Applications, Inc., Hampton, VA 23666, USA^{2}NASA Langley Research Centre, Hampton, VA 23666, USA

^{1}Science Systems and Applications, Inc., Hampton, VA 23666, USA^{2}NASA Langley Research Centre, Hampton, VA 23666, USA

Abstract

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This study applies fuzzy *k*-means (FKM) cluster analyses to a
subset of the parameters reported in the CALIPSO lidar level 2 data products
in order to classify the layers detected as either clouds or aerosols. The
results obtained are used to assess the reliability of the cloud–aerosol
discrimination (CAD) scores reported in the version 4.1 release of the
CALIPSO data products. FKM is an unsupervised learning algorithm, whereas
the CALIPSO operational CAD algorithm (COCA) takes a highly supervised
approach. Despite these substantial computational and architectural
differences, our statistical analyses show that the FKM classifications
agree with the COCA classifications for more than 94 % of the cases in
the troposphere. This high degree of similarity is achieved because the
lidar-measured signatures of the majority of the clouds and the aerosols are
naturally distinct, and hence objective methods can independently and
effectively separate the two classes in most cases. Classification
differences most often occur in complex scenes (e.g., evaporating water
cloud filaments embedded in dense aerosol) or when observing diffuse
features that occur only intermittently (e.g., volcanic ash in the tropical
tropopause layer). The two methods examined in this study establish overall
classification correctness boundaries due to their differing algorithm
uncertainties. In addition to comparing the outputs from the two algorithms,
analysis of sampling, data training, performance measurements, fuzzy linear
discriminants, defuzzification, error propagation, and key parameters in
feature type discrimination with the FKM method are further discussed in
order to better understand the utility and limits of the application of
clustering algorithms to space lidar measurements. In general, we find that
both FKM and COCA classification uncertainties are only minimally affected
by noise in the CALIPSO measurements, though both algorithms can be
challenged by especially complex scenes containing mixtures of discrete
layer types. Our analysis results show that attenuated backscatter and
color ratio are the driving factors that separate water clouds from
aerosols; backscatter intensity, depolarization, and mid-layer altitude are
most useful in discriminating between aerosols and ice clouds; and the joint
distribution of backscatter intensity and depolarization ratio is critically
important for distinguishing ice clouds from water clouds.

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Zeng, S., Vaughan, M., Liu, Z., Trepte, C., Kar, J., Omar, A., Winker, D., Lucker, P., Hu, Y., Getzewich, B., and Avery, M.: Application of high-dimensional fuzzy *k*-means cluster analysis to CALIOP/CALIPSO version 4.1 cloud–aerosol discrimination, Atmos. Meas. Tech., 12, 2261-2285, https://doi.org/10.5194/amt-12-2261-2019, 2019.

1 Introduction

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The Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observation (CALIPSO) mission has been developed through a close and ongoing collaboration between NASA Langley Research Center (LaRC) and the French space agency, Centre National D'Études Spatiales (CNES) (Winker et al., 2010). This mission provides unique measurements to improve our understanding of global radiative effects of clouds and aerosols in the Earth's climate system. The CALIPSO satellite was launched in April 2006, as a part of the A-Train constellation (Stephens et al., 2018). The availability of continuous, vertically resolved measurements of the Earth's atmosphere at a global scale leads to great improvements in understanding both atmospheric observations and climate models (Konsta et al., 2013; Chepfer et al., 2008).

The Cloud-Aerosol Lidar with Orthogonal Polarization (CALIOP), aboard CALIPSO, is the first satellite-borne polarization-sensitive lidar that specifically measures the vertical distribution of clouds and aerosols along with their optical and geometrical properties. The level 1 CALIOP data products report vertically resolved total atmospheric backscatter intensity at both 532 and 1064 nm, and the component of the 532 nm backscatter that is polarized perpendicular to the laser polarization plane. The level 2 cloud and aerosol products are retrieved from the level 1 data and separately stored into two different file types: the cloud, aerosol, and merged layer product files (CLay, ALay, and MLay, respectively) and the cloud and aerosol profile product files (CPro and APro). The profile data are generated at 5 km horizontal resolution for both clouds and aerosols, with vertical resolutions of 60 m from −0.5 to 20.2 km, and 180 m from 20.2 to 30 km. The layer data are generated at 5 km horizontal resolution for aerosols and at three different horizontal resolutions for clouds (1∕3, 1 and 5 km). The layer products consist of a sequence of column descriptors (e.g., latitude, longitude, time) that provide information about the vertical column of atmosphere being evaluated. Each set of column descriptors is associated with a variable number of layer descriptors that report the spatial and optical properties of each layer detected in the column.

The CALIOP level 2 processing system is composed of three modules, which have the general functions of detecting layers, classifying the layers, and performing extinction retrievals. These three modules are the selective iterated boundary locator (SIBYL), the scene classifier algorithm (SCA), and the hybrid extinction retrieval algorithm (HERA) (Winker et al., 2009). The level 2 lidar processing begins with the SIBYL module that operates on a sequence of scenes consisting of segments of level 1 data covering 80 km in along-track distance. The module averages these profiles to horizontal resolutions of 5, 20, and 80 km, respectively, and detects features at each of these resolutions. Those features detected at 5 km are further inspected to determine if they can also be detected at finer spatial scales (Vaughan et al., 2009). The SCA is composed of three main submodules: the cloud and aerosol discrimination (CAD) algorithm (Liu et al., 2004, 2009, 2019), the aerosol subtyping algorithm (Omar et al., 2009; Kim et al., 2018), and the cloud ice–water phase discrimination algorithm (Hu et al., 2009; Avery et al., 2019). Profiles of particulate (i.e., cloud or aerosol) extinction and backscatter coefficients and estimates of layer optical depths are retrieved for all feature types by the HERA module.

Clouds and aerosols modulate the Earth's radiation balance in different ways, depending on their composition and spatial and temporal distributions, and thus being able to accurately discriminate between them using global satellite measurements is critical for better understanding trends in global climate change (Trenberth et al., 2009). The CALIOP operational CAD algorithm (COCA) uses a family of multi-dimensional probability density functions (PDFs) to distinguish between clouds and aerosols (Liu et al., 2004, 2009, 2019). Using a larger number of layer attributes (i.e., higher-dimension PDFs) generally yields increasingly accurate cloud and aerosol discrimination. While both V3 and V4 COCA algorithms use the same five attributes to derive their classifications, substantial improvements have been made in V4 due to much improved calibration, especially at 1064 nm (Liu et al., 2019; Vaughan et al., 2019). The V4 PDFs have been rebuilt to better discriminate dense dust over the Taklamakan Desert, lofted dust over Siberia and the American Arctic regions, and high-altitude smoke and volcanic aerosol. Also, the application of the V4 PDFs has been extended, and they are now used to discriminate between clouds and aerosols in the stratosphere and to features detected at single-shot resolution (333 m) in the mid-to-lower troposphere.

CALIPSO has been delivering separate cloud and aerosol data products
throughout its 12+-year lifetime, and the reliable segregation of these
products clearly depends on the accuracy of the COCA. However, to the best
of our knowledge, no traditional validation study of the CALIOP CAD results
has been published in the peer-reviewed literature. Traditional validation
studies typically compare coincident measurements of identical phenomena
acquired by previously validated and well-established instruments to the
measurements acquired by the instrument being validated. For example,
radiometric calibration of the CALIOP attenuated backscatter profiles have
been extensively validated using ground-based Raman lidars (Mamouri et al.,
2009; Mona et al., 2009) and airborne high spectral resolution lidars (HSRL)
(Kar et al., 2018; Getzewich et al., 2018). Furthermore, CALIOP level 2
products have also been thoroughly validated: cirrus cloud heights and
extinction coefficients have been validated using measurements by Raman
lidars (Thorsen et al., 2011), Cloud Physics Lidar (CPL) measurements (Yorks
et., 2011; Hlavka et al., 2012), and in situ observations (Mioche et al.,
2010); CALIOP aerosol typing has been assessed by HSRL measurements (Burton
et al., 2013) and Aerosol Robotic Network (AERONET) retrievals (Mielonen et
al., 2009); and CALIOP aerosol optical depth estimates have been validated
using HSRL measurements (Rogers et al., 2014), Raman measurements (Tesche et
al., 2013), AERONET measurements (Schuster et al., 2012; Omar et al., 2013),
and Moderate Resolution Imaging Spectroradiometer (MODIS) retrievals
(Redemann et al., 2012). These level 2 validation studies implicitly depend
on the assumption that the COCA classifications are essentially correct;
however, this fundamental assumption has yet to be verified. This paper is,
therefore, a first step in an ongoing process of verifying and validating
the outputs of the CALIOP operational CAD algorithm. But unlike traditional
validation studies in which coincident measurements are compared, this study
will compare the outputs of two wholly different classification schemes
applied to the same measured input data. Clearly, one of these two schemes is
COCA. The other is the venerable fuzzy *k*-means (FKM) clustering algorithm,
which has a long history of use in classifying features found in satellite
imagery (Harr and Elsberry, 1995; Metternicht, 1999; Burrough et al., 2001;
Olthof and Latifovic, 2007; Jabari and Zhang, 2013).

The rationale for comparing algorithm outputs rather than measurements is twofold. First, no suitable set of coincident observations is currently available for use in a global-scale validation study. The spatial and temporal coincidence of ground-based and airborne measurements is extremely limited, and thus any validation exercise would require assumptions about the compositional persistence of features being compared. (Paradoxically, these are precisely the sorts of assumptions that should be obviated by well-designed validation studies.) Coincident A-Train measurements can be used in simple cases (Stubenrauch et al., 2013) but have little to offer in the complex scenes where clouds and aerosols intermingle; e.g., passive sensors cannot provide comparative information in multi-layer scenes or at cloud–aerosol boundaries, and the CloudSat radar is only sensitive to large particles and thus cannot help to distinguish between scattering targets that it cannot detect (e.g., lofted dust and thin cirrus). Second, COCA is a highly supervised classification scheme whose decision-making prowess depends on human-specified PDFs. FKM, on the other hand, is an unsupervised learning algorithm that, after suitable training, delivers classifications based on the inherent structure found in the data. The results obtained from the two different algorithms will help us better understand global cloud and aerosol distributions, which is important for all the users of space lidar (e.g., atmospheric scientists, weather and climate modelers, instrument developers). The flexibility of the FKM approach can help determine which individual parameters are most influential in discriminating clouds from aerosols and help evaluate the degree of improvement to be expected if/when new observational dimensions are added to the COCA PDFs.

Our paper is structured as follows. Section 2 briefly reviews the fundamentals of the COCA PDFs and their application to the CALIOP measurements. Section 3 provides an overview of the FKM algorithm and describes how we have adapted it for use in the CALIOP cloud–aerosol discrimination task. Section 4 compares the FKM classifications to the V3 and V4 COCA results. These comparisons, which are made for both individual cases and statistical aggregates, are designed to assess the accuracy of the COCA algorithm in general and to quantify changes in performance that can be attributed to the algorithm refinements incorporated in V4 (Liu et al., 2019). Various FKM performance metrics are described in Sect. 5, including error propagation, key parameter analysis, fuzzy discriminant analysis and principle component analysis. Conclusions and perspectives are given in Sect. 6.

2 CALIOP CAD PDF construction

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The CALIOP operational CAD algorithm uses manually derived,
multi-dimensional PDFs together with a statistical discrimination function
to distinguish between clouds and aerosols. Given a standard set of lidar
measurements (*X*_{1}, *X*_{2}, … *X*_{m}), separate
multi-dimensional PDFs are constructed for clouds (*P*_{cloud} (*X*_{1},
*X*_{2}, … *X*_{m})) and aerosols (*P*_{aerosol} (*X*_{1},
*X*_{2}, … *X*_{m})). Discrimination between clouds and aerosols
for previously unclassified layers is then determined using

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}f\left({X}_{\mathrm{1}},{X}_{\mathrm{2}},\mathrm{\dots},{X}_{m}\right)\\ \text{(1)}& {\displaystyle}& {\displaystyle}\phantom{\rule{1em}{0ex}}={\displaystyle \frac{{\mathit{P}}_{\mathrm{cloud}}\left({X}_{\mathrm{1}},{X}_{\mathrm{2}},\mathrm{\dots},{X}_{m}\right)-{\mathit{P}}_{\mathrm{aerosol}}\left({X}_{\mathrm{1}},{X}_{\mathrm{2}},\mathrm{\dots},{X}_{m}\right)\phantom{\rule{0.125em}{0ex}}k}{{\mathit{P}}_{\mathrm{cloud}}\left({X}_{\mathrm{1}},{X}_{\mathrm{2}},\mathrm{\dots},{X}_{m}\right)+{\mathit{P}}_{\mathrm{aerosol}}\left({X}_{\mathrm{1}},{X}_{\mathrm{2}},\mathrm{\dots},{X}_{m}\right)\phantom{\rule{0.125em}{0ex}}k}}..\end{array}$$

The function *f* is a normalized differential probability, with values that range
between −1 and 1, and *k* is a scaling factor that is related to the ratio of the
numbers of aerosol layers and cloud layers used to develop the PDFs (Liu et
al., 2009, 2019). Within the CALIOP level 2 data products, a
percentile (integer) value of 100×*f*, ranging from −100 to 100, is
reported as the “CAD score” characterizing each feature. Aerosol CAD
scores range from −100 to 0, and cloud CAD scores range from 0 to 100.
Because the nature of clouds is quite different from aerosols, most clouds
and aerosols can be distinguished unambiguously. Transition regions where
clouds are embedded in aerosols, volcanic ash injected into the upper
troposphere, and optically thick, strongly scattering aerosols at relatively
high altitudes (e.g., haboobs) can still present significant discrimination
challenges, but these cases occur relatively infrequently.

The initial version of COCA used only three layer attributes: layer mean
attenuated backscatter at 532 nm, $\langle {\mathit{\beta}}_{\mathrm{532}}^{\prime}\rangle $, layer-integrated attenuated backscatter color ratio,
${\mathit{\chi}}^{\prime}=\langle {\mathit{\beta}}_{\mathrm{1064}}^{\prime}\rangle /\langle {\mathit{\beta}}_{\mathrm{532}}^{\prime}\rangle $, and mid-layer altitude,
*z*_{mid}. Since then, the algorithm has been incrementally improved, and
beginning in V3 the COCA PDFs were expanded to five dimensions (5-D) by
adding layer-integrated 532 nm volume depolarization, *δ*_{v}, and
the latitude of the horizontal midpoint of the layer (Liu et al., 2019).
Within the CALIPSO analysis software, these PDFs are implemented as 5-D
arrays that function as look-up tables. However, while the V3 and V4
algorithms both use five independent measurements, the numerical values in
the underlying PDFs are significantly different. The V3 PDFs were rendered
obsolete by extensive changes to the V4 calibration algorithms (Kar et al.,
2018; Getzewich et al., 2018; Vaughan et al., 2019) that required revising a
number of the $\langle {\mathit{\beta}}_{\mathrm{532}}^{\prime}\rangle $ probabilities
and a global recalculation of the color ratio probabilities. As a result,
the V4 CAD algorithm can now be applied in the stratosphere and to layers
detected at single-shot resolution and has greatly improved performance when
identifying dense dust over the Taklamakan Desert, lofted dust over Siberian
and American Arctic regions, and high-altitude aerosols in the upper
troposphere and lower stratosphere (Liu et al., 2019).

3 Fuzzy *k*-means cluster analysis

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Cluster analysis is a useful statistical tool to group data into several
categories and has been successfully applied to satellite observations to
discriminate among different features of interest (Key et al., 1989; Kubat
et al., 1998; Omar et al., 2005; Zhang et al., 2007; Usman, 2013; Luo et al.,
2017; Gharibzadeh et al., 2018). There are many different types of
clustering methods, such as connectivity-based, centroid-based,
density-based, and distribution clustering, and these are typically trained
using either supervised or unsupervised learning techniques. In this paper,
we focus on a centroid-based, unsupervised learning approach known as the
fuzzy *k*-means (FKM) method. As the name implies, classification ambiguities
are expressed in terms of fuzzy logic (i.e., as opposed to “crisp”/binary
logic), and thus every point processed by the clustering algorithm is
assigned some degree of membership in all categories, rather than belonging
solely to just one category. FKM membership values range from 0 to 1 and
thus are comparable to the operational CAD scores. In addition, the shapes
and density distributions of multi-dimensional observations of clouds and
aerosols from lidar are well suited for the centroid-based clustering technique
used by the FKM classification method. With the exception of latitude, our
FKM implementation uses the same inputs as COCA; i.e., $\langle {\mathit{\beta}}_{\mathrm{532}}^{\prime}\rangle $, *χ*^{′}, *δ*_{v}, and
*z*_{mid}. We make this choice because clouds and aerosols show distinct
centers in the $\langle {\mathit{\beta}}_{\mathrm{532}}^{\prime}\rangle $, *δ*_{v}, *χ*^{′}, *z*_{mid} attribute space, whereas adding
latitude degrades the separation between cluster centers and adds
significantly to class overlap. A key parameter analysis (described in Sect. 5) demonstrates that latitude does not provide intrinsic information that
helps to distinguish between aerosols and cloud, nor does it improve the
reliability of the cluster membership values (e.g., Wilks' lambda, a measure
of the difference between classes also introduced in Sect. 5., deteriorates
from ∼0.2 to ∼0.5). However, for probabilistic
systems (e.g., COCA), latitude can be useful, simply because some feature
types are more likely than others to occur within specific latitude–altitude
bands (e.g., at altitudes of 9–11 km, significant aerosol loading is much
more likely at 45^{∘} N than at 60^{∘} S).

Given a set of observations ** X**= (

$$\begin{array}{}\text{(2)}& {\displaystyle}& {\displaystyle}{m}_{ij}\in \left[\mathrm{0},\mathrm{1}\right],\phantom{\rule{0.125em}{0ex}}i=\mathrm{1}\mathrm{\dots}n,\phantom{\rule{0.125em}{0ex}}j=\mathrm{1}\mathrm{\dots}k\text{(3)}& {\displaystyle}& {\displaystyle}\sum _{j=\mathrm{1}}^{k}{m}_{ij}=\mathrm{1},\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}i=\mathrm{1}\mathrm{\dots}n,\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\text{and}\text{(4)}& {\displaystyle}& {\displaystyle}\sum _{i=\mathrm{1}}^{n}{m}_{ij}>\mathrm{0},\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}j=\mathrm{1}\mathrm{\dots}k.\end{array}$$

To determine the best solution, based on minimization of the WCSS, a classic
objective function, *J*, is built so that the best solution is the one that
minimizes *J* (Bezdek, 1981, 1984; McBratney and Moore, 1985). The
functional form of *J* is

$$\begin{array}{}\text{(5)}& J\left(\mathbf{M},\mathbf{C}\right)=\sum _{i=\mathrm{1}}^{n}\sum _{j=\mathrm{1}}^{k}{m}_{ij}^{\mathit{\varphi}}\phantom{\rule{0.125em}{0ex}}{d}_{ij}^{\mathrm{2}}\left({x}_{il},{c}_{jl}\right),\end{array}$$

where **C** (*c*_{jl}; *j*=1, …, *k*; *l*=1, …, *p*) is a matrix of class centers,
and *d*^{2} (*x*_{il}, *c*_{jl}) is the squared distance between individual
*x*_{il} and class center *c*_{jl} according to a chosen
definition of distance (e.g., the Mahalanobis distance; see Sect. 2.3). The
objective function is the squared error from class centers weighted by the
*ϕ*th power (fuzzy weighting exponent) of the membership values.
For the least meaningful value, *ϕ*=1, *J* minimizes only at crisp
partitions (the memberships converge to either 0 or 1), with no overlap
between cluster boundaries. Increasing the value of *ϕ* tends to degrade
memberships towards fuzzier states where there are more overlaps between the
boundaries of clusters. For a specified value of *ϕ*, minimization of
objective function *J* optimizes the solutions for the membership matrix **M** and
its associated centroid matrix **C** (Bezdek, 1981; McBratney and deGruijter,
1992; Minasny and McBratney, 2002). Class centers are the averages of the
individual samples weighted by their class membership values raised to the
*ϕ*th power (Eq. 6). The membership (*m*_{ij}) of an individual
belonging to a class is the distance between the individual and the class
center divided by the sum of the distances between the individual and the
centers of all classes (Eq. 7), or

$$\begin{array}{}\text{(6)}& {\displaystyle}& {\displaystyle}{c}_{jl}={\displaystyle \frac{\sum _{j=\mathrm{1}}^{n}{m}_{ij}^{\mathit{\varphi}}\phantom{\rule{0.125em}{0ex}}{x}_{ij}}{\sum _{i=\mathrm{1}}^{n}{m}_{ij}^{\mathit{\varphi}}}},\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}j=\mathrm{1},\mathrm{2}\mathrm{\dots}k,\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}l=\mathrm{1},\mathrm{2}\mathrm{\dots}p,\text{(7)}& {\displaystyle}& {\displaystyle}{m}_{jl}={\displaystyle \frac{{d}_{ij}^{-\mathrm{2}/\left(\mathit{\varphi}-\mathrm{1}\right)}}{\sum _{j=\mathrm{1}}^{k}{d}_{ij}^{-\mathrm{2}/\left(\mathit{\varphi}-\mathrm{1}\right)}}},\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}i=\mathrm{1},\mathrm{2}\mathrm{\dots}n,\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}j=\mathrm{1},\mathrm{2}\mathrm{\dots}k,\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\text{and}\end{array}$$

To obtain centroid (Eq. 6) and membership (Eq. 7) solutions, Picard
iterations (Bezdek et al., 1984) are applied until the centers or
memberships are constant to within some small value (see the algorithm
flowchart in Fig. 1). We first initialize the memberships as random values
using a uniform distribution that satisfies all conditions given by
Eqs. (2), (3) and (4). We then calculate class centers and recalculate
memberships according to the new centers. If the new memberships do not
change compared to the old ones (or change only within a small difference
*ε*), the clustering process ends. Otherwise, we recalculate the
new centers and new memberships. If the algorithm does not converge after a
fixed number of iterations, the procedure is reinitiated using newly (and
again randomly) specified initial cluster centers. This process repeats
until the algorithm converges to a point where the relative change in the
objective function (calculated from Eq. 5, which quantifies the changes in
both the memberships and centers) is less than *ε* (0.001) and
saves the best memberships and centers that result from the optimum random
initiation corresponding to the least objective function.

Before running the FKM code (from Minasny and McBratney, 2002), we prepared our data by sampling, training, and filtering (Sect. 2.1 and 2.2). We also selected a reasonable method to calculate the distance between individuals and centers (Sect. 2.3) and determined optimal values for class number and fuzzy exponent (Sect. 2.4). Note the FKM method is directly applied to data to get membership instead of building PDF as in operational algorithm.

As mentioned above, four level 2 parameters are used for our cluster
analysis: layer mean attenuated backscatter at 532 nm, $\langle {\mathit{\beta}}_{\mathrm{532}}^{\prime}\rangle $, layer-integrated volume depolarization
ratio at 532 nm, *δ*_{v}, total attenuated backscatter color ratio,
*χ*^{′}, and mid-layer altitude, *z*_{mid}. The selection of four
dimensions is based on many previous studies (e.g., Liu et al., 2004, 2009;
Hu et al., 2009; Omar et al., 2009; Burton et al., 2013), which show that
clouds, aerosols, and their subtypes are quite different based on these
observations. $\langle {\mathit{\beta}}_{\mathrm{532}}^{\prime}\rangle $, *δ*_{v}, and *χ*^{′} are the fundamental lidar-derived optical
properties that form the basis for our discrimination scheme. We also
include altitude, as the joint distributions of altitude with the various
lidar-derived optical properties have proven to be highly effective in identifying
different feature types.

In this study, we apply FKM at a global scale. For any given region, results derived from a localized cluster analysis will likely give us better classifications compared to the results from a global-scale analysis, but investigating and/or characterizing these differences lies well beyond the scope of this study. The data sample size also strongly influences the clustering results. For example, clustering into two classes with a full complement of CALIPSO data could identify clear and “not clear” scenes. If clear scenes are excluded, clustering could separate clouds and aerosols. If only clear scenes are included, clustering could possibly provide a means of identifying different surface types. With only cloudy data, clustering could be used to derive thermodynamic phase classification. With only aerosol data, clustering is actually aerosol subtyping. With only liquid cloud data, clustering could separate cumulus and stratocumulus. So, the size and composition of the dataset is very important for our analysis, which strongly depends on the objective of the classification.

To extrapolate the classification of identifiable elements using FKM from a
small subset to a broader population, we identify an appropriate training
dataset from which the classifications can be derived (Burrough et al.,
2000). This training data should be representative of the broader sample for
which the classification will be implemented (i.e., both must span similar
domains). To ensure the selection of an appropriate training dataset, the
shapes of the PDFs of the relevant parameters derived from any proposed
training set should closely match the shapes of the corresponding parameter
PDFs derived from the global long-term dataset. Data from the month of
January 2008 are used to determine the optimal number of classes (*k*) and fuzzy
exponent (*ϕ*) required for classification and optimal values of the
performance parameters, and to calculate class centroids for interpretation
of similarities and differences between classes. To avoid errors due to
small sample sizes, we used the same month of global observations (January
2008) to do the subsequent comparisons with COCA results.

Figure 2 shows approximate probability density functions (computed by
normalizing the sum of the occurrence frequencies to 1) for different lidar
observables for liquid water clouds, ice
clouds composed of either randomly or horizontally oriented ice crystals, and aerosols during all of January 2008. Liquid water clouds have
the largest $\langle {\mathit{\beta}}_{\mathrm{532}}^{\prime}\rangle $ and *χ*^{′} values compared with other species. Aerosols generally have the
smallest *χ*^{′}, *δ*_{v}, and $\langle {\mathit{\beta}}_{\mathrm{532}}^{\prime}\rangle $, and ice clouds have the largest *δ*_{v}
compared with the other two species. There is overlap between species, but
these three parameters are still sufficient to separate aerosols and
different phases of cloud in most cases. Figure 2d, e, and
f are from a single half orbit (2008-09-06T01-35-29ZN) of
observations. The PDFs of one half orbit and 1 month of observations
appear to agree very well, which means that focused feature clustering
studies that use the FKM method can be applied to a small sample such as
one-half orbit of observations and not cause significant biases to the
standard full dataset.

We filtered the training data to eliminate outliers in the $\langle {\mathit{\beta}}_{\mathrm{532}}^{\prime}\rangle $, *δ*_{v}, and *χ*^{′}
measurements that were physically unrealistic (i.e., either too high or too
low). Eliminating these extreme values speeds up the processing, and the
training algorithm converges more rapidly. The selected filter thresholds
retain more than 98 % of all features within the original dataset. A
summary of the thresholds is given in Table 1. The selection of these
thresholds is based on the PDFs shown in Fig. 2.

The distances between attributes can be calculated in different ways (e.g.,
Euclidean distance, diagonal distance and Mahalanobis distance). According
to a study by Gorsevski et al. (2003), we should apply the Euclidean
distance to uncorrelated variables on the same scale when attributes are
independent and the clusters are spherically shaped clouds. The diagonal
distance is also insensitive to statistically dependent variables but
clusters are not required to have spherically shaped clouds. The Mahalanobis
distance can be used for correlated variables on the same or different
scales and when the clusters are ellipsoid-shaped clouds. The Mahalanobis
distance (*d*_{ij}) of an observation *i* from a set of observations
(*x*_{il}) with centers *c*_{jl} (*x*_{il}-*c*_{jl} is an
*l*-dimensional vector) is defined in Eq. (8) (Mahalanobis, 1936):

$$\begin{array}{ll}{\displaystyle}{d}_{ij}^{\mathrm{2}}& {\displaystyle}={\left({x}_{il}-{c}_{jl}\right)}^{T}\phantom{\rule{0.125em}{0ex}}{\mathbf{S}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}\left({x}_{il}-{c}_{jl}\right),\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}i=\mathrm{1},\mathrm{2}\mathrm{\dots}n,\\ \text{(8)}& {\displaystyle}& {\displaystyle}j=\mathrm{1},\mathrm{2}\mathrm{\dots}k,\phantom{\rule{0.125em}{0ex}}l=\mathrm{1},\mathrm{2}\mathrm{\dots}p.\end{array}$$

**S**^{−1} (an *l*×*l* matrix) is the inverse of the covariance matrix
of the observations. Note superscript *T* indicates that the vector should be
transposed. If the covariance matrix is a diagonal matrix, the Mahalanobis
distance calculation returns the normalized Euclidean distance. In this work,
we use the Mahalanobis distance specifically because the three lidar
observables used both in FKM and COCA are not independent. Each is a sum (or
mean) of the measured backscatter signal over some altitude range, with the
relationships between them given as follows:

$$\begin{array}{}\text{(9a)}& {\displaystyle}& {\displaystyle}\u2329{\mathit{\beta}}_{\mathrm{532}}^{\prime}\u232a={\displaystyle \frac{\mathrm{1}}{N}}\sum _{n=\mathrm{1}}^{N}{\mathit{\beta}}_{\mathrm{532},\parallel}^{\prime}\left({z}_{n}\right)+{\mathit{\beta}}_{\mathrm{532},\perp}^{\prime}\left({z}_{n}\right),\text{(9b)}& {\displaystyle}& {\displaystyle}{\mathit{\delta}}_{\mathrm{v}}={\displaystyle \frac{\sum _{n=\mathrm{1}}^{N}{\mathit{\beta}}_{\mathrm{532},\perp}^{\prime}\left({z}_{n}\right)}{\sum _{n=\mathrm{1}}^{N}{\mathit{\beta}}_{\mathrm{532},\parallel}^{\prime}\left({z}_{n}\right)}},\text{(9c)}& {\displaystyle}& {\displaystyle}\text{and}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}{\mathit{\chi}}^{\prime}={\displaystyle \frac{\u2329{\mathit{\beta}}_{\mathrm{1064}}^{\prime}\u232a}{\langle {\mathit{\beta}}_{\mathrm{532}}^{\prime}\rangle}}={\displaystyle \frac{\sum _{n=\mathrm{1}}^{N}{\mathit{\beta}}_{\mathrm{1064}}^{\prime}\left({z}_{n}\right)}{\sum _{n=\mathrm{1}}^{N}{\mathit{\beta}}_{\mathrm{532}}^{\prime}\left({z}_{n}\right)}}.\end{array}$$

In these expressions, the subscripts ∥ and ⊥ represent contributions from the 532 nm parallel and perpendicular channels, respectively. Note in particular that the signals measured in the 532 nm parallel channel contribute to all three quantities.

The selection of an optimal number of classes *k* (1 < *k* < *n*)
and degree of fuzziness *ϕ* (*ϕ* > 1) has been discussed
in many previous studies (Bezdek, 1981; Roubens, 1982; McBratney and Moore,
1985; Gorsevski, 2003). The number of classes specified should be meaningful
in reality and the partitioning of each class should be stable. For each
generated classification, analyses need to be performed to validate the
results. Among different validation functions, the fuzzy performance index
(FPI) and the modified partition entropy (MPE) are considered two of the
most useful indices among seven examined by Roubens (1982) to evaluate the
effects of varying class number. The FPI is defined as in Eq. (10), where *F* is
the partition coefficient calculated from Eq. (11). The MPE is defined as in
Eq. (12), with the entropy function (*H*) calculated from Eq. (13).

The ideal number of continuous and structured classes (*k*) can be established
by simultaneously minimizing both FPI and MPE. For the fuzziness exponent,
if the value of *ϕ* is too low, the classes become more discrete and the
membership values either approach 0 or 1. But if *ϕ* is too high, the
classes will not provide useful discrimination among samples and
classification calculations may fail to converge. McBratney and Moore (1985)
suggested that the objective function (Eq. 14, Bezdek, 1981) decreases with
increasing of both fuzzy exponent (*ϕ*) and the number of classes (*k*).
They plotted a series of objective functions versus the fuzzy exponent
(*ϕ*) for a given class where the best value of *ϕ* for that class
is at the first maximum of objective function curves (Odeh et al., 1992a;
McBratney and Moore, 1985; Triantafilis et al., 2003). Therefore, choosing an optimal combination of
class number (*k*) and fuzzy exponent (*ϕ*) is established on the basis of
minimizing both values of FPI and MPE and the least maximum of the objective
function, where

$$\begin{array}{}\text{(10)}& {\displaystyle}& {\displaystyle}\mathrm{FPI}=\mathrm{1}-{\displaystyle \frac{k\times F-\mathrm{1}}{k-\mathrm{1}}},\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\text{and}\text{(11)}& {\displaystyle}& {\displaystyle}F={\displaystyle \frac{\mathrm{1}}{n}}\sum _{i=\mathrm{1}}^{n}\sum _{j=\mathrm{1}}^{k}{m}_{ij}^{\mathrm{2}};\text{(12)}& {\displaystyle}& {\displaystyle}\mathrm{MPE}={\displaystyle \frac{H}{\mathrm{log}\phantom{\rule{0.125em}{0ex}}k}},\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\text{and}\text{(13)}& {\displaystyle}& {\displaystyle}H={\displaystyle \frac{\mathrm{1}}{n}}\sum _{i=\mathrm{1}}^{n}\sum _{j=\mathrm{1}}^{k}{m}_{ij}\times \mathrm{log}\left({m}_{ij}\right);\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\text{and}\text{(14)}& {\displaystyle}& {\displaystyle \frac{\partial \mathbf{J}\left(\mathbf{M},\mathbf{C}\right)}{\partial \mathit{\varphi}}}=\sum _{i=\mathrm{1}}^{n}\sum _{j=\mathrm{1}}^{k}{m}_{ij}^{\mathit{\varphi}}\mathrm{log}\left({m}_{ij}\right)\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}{d}_{ij}^{\mathrm{2}}.\end{array}$$

Using 1 month of layer optical properties reported in the CALIOP level 2
merged layer products, we created Fig. 3 to determine optimal values for *k*
and *ϕ*. From this figure, we conclude that the ideal number classes for
CALIOP layer classification is either three or four, with corresponding fuzzy
exponents equal to 1.4 or 1.6 (we use 1.4 for the analyses in this paper).
Before exploring the clustering results to see what each class represents,
we can immediately confirm that using three classes would be physically
meaningful (i.e., these three classes may be aerosols, liquid water clouds, and
ice clouds). Similarly, two classes could represent aerosols and clouds. In
the following study, we will choose *k* equal to 2 or 3 and *ϕ* equal to
1.4.

4 Cluster results and comparison with V3 and V4 data

Back to toptop
According to Liu et al. (2009), the CAD score for any layer is the difference between the probability of being a cloud and the probability of being an aerosol (Eq. 15). We calculate the FKM CAD score in a similar way, where the COCA probabilities are replaced with FKM membership values. For the three-class FKM analyses, the cloud membership value is the sum of memberships of ice and water clouds (two classes). The FKM CAD score is found using

$$\begin{array}{}\text{(15)}& {\mathrm{CAD}}_{\mathrm{FKM}}={\displaystyle \frac{{\mathit{M}}_{\mathrm{cloud}}-{\mathit{M}}_{\mathrm{aerosol}}}{{\mathit{M}}_{\mathrm{cloud}}+{\mathit{M}}_{\mathrm{aerosol}}}}\times \mathrm{100}.\end{array}$$

Figure 4 compares the operational V3 and V4 CAD products and our CAD_{FKM}
classifications for a single nighttime orbit segment (6 September 2008,
beginning at 01:35:29 GMT). Generally speaking, CAD_{FKM} values from both the
two-class and three-class analyses are quite similar to both the V3 and V4 COCA
values. When COCA CAD scores are positive (namely clouds, shown in whitish
colors in Fig. 4) in V3 and V4, the two-class and three-class CAD_{FKM} values
are also positive. Likewise, when COCA CAD scores are negative (namely
aerosols, yellowish colors in Fig. 4), the two-class and three-class CAD_{FKM}
values are also negative. Furthermore, the particular orbit selected here
includes the observations of a plume of high, dense smoke lofted over low
water clouds (latitudes between 0^{∘} and 20^{∘} S, shown
within the red oval in Fig. 4a). For these water clouds beneath dense smoke,
both the V3 operational CAD and the two-class CAD_{FKM} label them as clouds
with low positive values. On the other hand, the V4 operational CAD and the
three-class CAD_{FKM} return higher values much closer to 100. The reasons for
these differences will be discussed in Sect. 5.2 and 5.4. Note too
that weakly scattering edges of cirrus clouds (hereafter, cirrus fringes)
beyond 69.6^{∘} S are misclassified as aerosols by both the two-class
and three-class CAD_{FKM} (Fig. 4c and d) but are correctly classified as
clouds by the operational V4 algorithms.

The differences between the FKM and V4 COCA classifications are most
prominent for layers detected in the stratosphere (i.e., layers rendered in
black in Fig. 4b). In the Northern Hemisphere, between 71.55
and 36.86^{∘} N, a diffuse, weakly scattering layer is intermittently
detected at altitudes between 12 and 18 km. This layer most likely
originated with the eruption of the Kasatochi volcano on 7 August 2008
(Krotkov et al., 2010). But while V4 COCA classifies these layers as
aerosols, both FKM methods identify them as clouds. In the Southern
Hemisphere, south of 69.6^{∘} S, a faint polar stratospheric layer is
detected continuously, with a mean base altitude at ∼11 km
and a mean top at ∼15 km. Once again, the V4 COCA classifies
this feature as a moderate- to high-confidence aerosol layer and the three-class
FKM classifies it as a high-confidence cloud. However, unlike the Northern
Hemisphere case, the two-class FKM identifies this as low-confidence aerosol.
Correctly classifying stratospheric features that occupy the “twilight
zone” between aerosols and highly tenuous clouds (Koren et al., 2007) is
likely to be difficult for unsupervised learning methods, due to the
extensive overlap in the available lidar observables for the two classes.
Class separation is typically (though not always) more distinct within the
troposphere.

The confusion index (CI) is a measure of the degree of class overlap or
uncertainty between classes (Burrough and McDonnell, 1998). In effect, it
measures how confidently each individual observation has been classified. CI
values are calculated from Eq. (16), where *m*_{max} denotes the biggest
membership value and *m*_{max−1} is the second biggest membership value
for each individual observation (*i*):

$$\begin{array}{}\text{(16)}& \mathrm{CI}=\left[\mathrm{1}-\left({m}_{{max}_{i}}-{m}_{{\left(max-\mathrm{1}\right)}_{i}}\right)\right].\end{array}$$

The CI value approaches zero when *m*_{max} is much larger than *m*_{max−1},
indicating that the observation is more likely to belong to one dominant
class. CI approaches 1 when *m*_{max} is almost equal to *m*_{max-1}. In
such cases, the difference between the dominant and subdominant classes is
negligible, which creates confusion in the classification of that particular
observation. Note the value (1 – CI) ×100 for the two-class FKM
algorithm is equivalent to the absolute value of the CAD_{FKM} score.

Figure 5 shows CI values for two-class and three-class CAD_{FKM} calculated for
all layers in the sample orbit. From the figure, we see that, in most cases,
the CI values are low for both the two-class and three-class CAD_{FKM}
classifications. The exceptions are stratospheric features (mostly near
polar regions), cloud fringes, high-altitude aerosols, and, for two-class
CAD_{FKM} only, the liquid water clouds beneath dense smoke. Low CI values
for the CAD_{FKM} classifications are analogous to high CAD scores
assigned by the operational CAD algorithm: both indicate high-confidence
classifications. Similarly, CAD_{FKM} classifications with high CI values
indicate low-confidence classifications where the observation has roughly
equal membership in two classes. For the liquid water clouds beneath dense
smoke, the membership values determined by the two-class CAD_{FKM} are
larger than 0.5. However, the three-class CAD_{FKM} results for these water
clouds have low CI values, indicating high-confidence classifications into
one dominant class, and suggesting that the separation between the aerosols
and low water clouds is better accomplished when three classes are used. For
cloud fringes, the CI values are high for both the two-class and three-class
CAD_{FKM}. According to the CAD_{FKM} results, cirrus fringes are
somewhat different from the neighboring portions of the cirrus layer, as
they also bear some similarity to the dust particles that are the
predominant sources of ice nuclei (DeMott et al., 2010).

In this subsection, we present statistical analyses of our results for all
of January 2008, followed by explorations of individual case studies in the
next subsection. We first compare the PDFs of the different lidar-derived optical
parameters used in the two-class and three-class CAD_{FKM} classifications to
the PDFs of those same parameters derived for the COCA classifications (Fig. 6). We also compare the spatial distribution patterns of the clouds and
aerosols identified by FKM and COCA (Fig. 7) and use confusion matrices to
quantify the similarity of the corresponding FKM and COCA classes (Table 2).

From Fig. 6, it is evident that the PDFs of $\langle {\mathit{\beta}}_{\mathrm{532}}^{\prime}\rangle $, *δ*_{v}, and *χ*^{′} that
characterize the clouds and aerosols determined by the FKM classifications
agree well with the PDFs from the V4 CAD classifications. Figure 6d, e,
and f compare the two-class CAD_{FKM} results to the operational algorithm.
In these figures, the PDFs of $\langle {\mathit{\beta}}_{\mathrm{532}}^{\prime}\rangle $ (Fig. 6d), *δ*_{v} (Fig. 6e), and *χ*^{′}
(Fig. 6f) of FKM class 1 (dashed blue lines) agree well with
those of V4 cloud PDFs (solid blue lines), while the PDFs of these different
parameters of FKM class 2 (dashed red lines) agree well with those of V4
aerosol (solid red lines) PDFs. Figure 6a, b, and c compare the three-class
CAD_{FKM} results to the operational algorithm. Once again, the
comparisons are quite good: the shapes of the PDFs of FKM class 1 (dashed blue)
agree well with the V4 water cloud (blue solid) PDFs, while the PDFs
of FKM class 2 (dashed red) and 3 (dashed green) individually agree well
with, respectively, the V4 ice cloud (red solid) and aerosol (green solid)
PDFs. The class means for $\langle {\mathit{\beta}}_{\mathrm{532}}^{\prime}\rangle $
are smallest for aerosols/class 3 (0.0034±0.0022
and 0.0041±0.0193 (km^{−1} sr^{−1}),
respectively) and slightly larger for ice clouds/class 2 (0.0075±0.0086 and 0.0062±0.0183 (km^{−1} sr^{−1}),
respectively). Water clouds/class 1 have the largest $\rangle {\mathit{\beta}}_{\mathrm{532}}^{\prime}\rangle $ mean values (0.0804±0.0526
(km^{−1} sr^{−1}) and 0.0850±0.0454 (km^{−1} sr^{−1}),
respectively). For *δ*_{v}, the largest mean values are found for
ice clouds/class 2, followed by water clouds/class 1 and then aerosol/class
3. Class mean *χ*^{′} is largest for water clouds/class 1 and
smallest for aerosols/class 3. These means and standard deviations are also
comparable between COCA and FKM classes.

Figure 7 compares the geographical (panels a–f) and zonally averaged (panels
g–l) distributions of two-class CAD_{FKM} occurrence frequencies to the COCA
cloud and aerosol occurrence frequencies for all data acquired during
January 2008. The spatial distributions of clouds and aerosols are quite
different. In January, clouds are mostly located in the storm tracks, to the
east of continents, over the Intertropical Convergence Zone (ITCZ) and in
polar regions. Aerosols are more often found over the Sahara, over the
subtropical oceans, and in south-central and east Asia (Fig. 7a–f). In the
zonal mean plots (Fig. 7g–l), cloud tops are
seen to extend up to the local tropopause, whereas aerosols are largely
confined to the boundary layer. The geographical and vertical distributions
of FKM class 1 are quite similar to the COCA V4 cloud distributions.
Likewise, the distributions of FKM class 2 closely resemble the COCA V4
aerosol distributions. Looking at the difference plots (right-hand column of
Fig. 7), some fairly large differences are seen in the polar regions, where
the composition and intermingling of clouds and aerosols are notably
different from other regions of the globe. Many of the layers observed in
the polar regions are spatially diffuse and optically thin, and thus occupy
the morphological twilight zone between clouds and aerosols (Koren et al.,
2007). Observationally based validation of the feature types in these
regions would likely require extensive in situ measurements coincident with
CALIPSO observations. Consequently, correctly interpreting the
classifications by the two algorithms in polar regions based on our
knowledge is too challenging to draw useful conclusions and lies well beyond
the scope of this work. Nevertheless, the PDFs and geographic analyses
presented here establish that, excluding the polar regions, the
cloud–aerosol discrimination derived using an unsupervised FKM method is
statistically consistent with the classifications produced by the
operational V4 CAD algorithm.

Above, we qualitatively show the operational classification algorithm agrees
well with FKM algorithm. To quantify the degree to which the different
methods agree with each other, we construct confusion matrices, which use
the January 2008 5 km merged layer data between 60^{∘} S and
60^{∘} N to calculate the concurrent frequency of cloud and aerosol
identifications made by the COCA and CAD_{FKM} algorithms. We summarize
the occurrence frequency statistics in Table 2. From the table, we find that
for our test month COCA V3 agrees with COCA V4 CAD for 96.6 % of the
cases. The agreements are around 90 % for the entire globe including
regions beyond 60^{∘} (not shown here). The FKM two-class and three-class
results agree with both V3 and V4 for more than 93 % of the cases. The
FKM results agree slightly better with V3 than with V4. All algorithms and
versions agree on cloud coverage of around 58 % to 66 % of the globe.
These values are well within typical cloud climatology estimates of 50 %
to 70 % (Stubenrauch et al., 2013). Compared to the two-class CAD_{FKM},
results from the three-class CAD_{FKM} agree somewhat better with the
classifications from both the V3 and V4 CAD algorithms. Consistent with
previous results in this paper, the three-class CAD_{FKM} appears better able
to separate clouds and aerosols than the two-class CAD_{FKM}. Figure 4
provides an additional example. For those water clouds beneath dense smoke,
the three-class CAD_{FKM} scores are substantially higher than both the
two-class CAD_{FKM} scores and the operation V3 CAD scores, indicating that
the three-class CAD_{FKM} algorithm correctly identifies these features with
much higher classification confidence. While the discrepancies between the
two techniques are pleasingly small, their root causes are still of some
interest. For example, we note that the FKM algorithm shows a slight bias
toward aerosols relative to the V4 COCA (a 2.4 % bias for the three-class FKM
versus a 1.5 % bias for the two-class FKM). At present, we speculate that
the bulk of these differences can be traced to the dichotomy between
supervised (COCA) and unsupervised (FKM) learning techniques. Given the scope
and quality of the data currently available for use by the COCA and FKM
methods, the correct classification of layers occupying the twilight zone
separating clouds and aerosols remains somewhat uncertain, and hence
different learning strategies are likely to come to different conclusions,
even when provided the same evidence. We also calculated the concurrent
occurrence frequencies for only those features with CI values less than 0.75
(or 0.5). When the data are restricted to only relatively high-confidence
classifications, the FKM results agree with V3 and V4 for better than 96 % (or 97 %) of the samples tested.

In this section, we investigate several of the challenging classification
cases that motivated the extensive changes made in COCA in the transition
from V3 to V4 (Liu et al., 2019). Comparisons are done for those cases
between different algorithms and different algorithm versions to see how
well each algorithm or version compares to “the truth” (i.e., as obtained
by expert judgments). In addition to the dense smoke over opaque water cloud
case shown in Fig. 4, the CAD_{FKM} algorithm, like the
operational CAD algorithm, can occasionally have difficulty correctly
identifying high-altitude smoke, dense dust, lofted dust, cirrus fringes,
polar stratospheric clouds (PSCs), and stratospheric volcanic ash (Figs. 8–10). We briefly review each of these cases below.

Two different dust cases are selected for this study (Fig. 8). The first
case examines nighttime measurements of a deep and sometimes extremely dense
dust plume in the Taklamakan Desert beginning at 20:15:32 UTC on 4 May 2008,
as shown in Fig. 8a–e. The second case investigates spatially diffuse
Asian dust lofted high into the atmosphere while being transported toward
the Arctic during a nighttime orbit segment beginning at 18:28:54 UTC on 1 March 2008, as shown in Fig. 8f–j. CAD classifications are color-coded as
follows: regions where no features were detected are shown in pure blue; V3
stratospheric features are shown in black; cirrus fringes are shown in pale
blue; aerosol-like features are shown using an orange-to-yellow spectrum,
with orange indicating higher confidence and yellow lower confidence; and
cloud-like features are rendered in grayscale, with brighter and whiter
hues indicating higher classification confidence. Dust layers in the
Taklamakan Desert exhibit high (532 nm) attenuated backscatter coefficients, high depolarization
ratios (not shown), and attenuated backscatter color ratios close to 1 (also
not shown). As seen between ∼44 and
∼40^{∘} N, layers with this combination of layer
optical properties are frequently misclassified as ice clouds in COCA V3
(Fig. 8b). However, in COCA V4, these same layers are much more likely to be
correctly classified as aerosol (Fig. 8c). The two-class and three-class
CAD_{FKM} classifications both agree with COCA V4 for the lofted aerosols
but misclassify the densest portions of the dust plume as a low-confidence
cloud. For the lofted Asian dust case shown in Fig. 8f–j, COCA V3
frequently misclassifies dust filaments as clouds, whereas COCA V4 correctly
identifies the vast majority as dust. (Note too that many more layers are
detected in V4 as a consequence of the changes made to the CALIOP 532 nm
calibration algorithms (Kar et al., 2018; Getzewich et al., 2018; Liu et
al., 2019).) The two-class and three-class CAD_{FKM} classifications are
essentially identical to those determined by COCA V4 but show higher
confidence values for the aerosol layers.

An unprecedented example of high-altitude smoke plumes was observed by
CALIPSO during the “Black Saturday” fires that started 7 February 2009,
quickly spread across the Australian state of Victoria, and eventually
lofted well into the stratosphere (de Laat et al., 2012). Figure 9 shows
extensive smoke layers at 10 km and higher on Monday, 10 February, between
20 and 40^{∘} S. In the V3 CALIOP data products,
stratospheric layers (i.e., layers with base altitudes above the local
tropopause) were not further classified as clouds or aerosols but instead
were designated as generic “stratospheric features” (Liu et al., 2019).
Consequently, COCA V3 misclassifies these smoke layers as clouds when their
base altitude is below the tropopause and as stratospheric features when the
base altitude is higher (Fig. 9b). On the other hand, the V4 CAD correctly
identifies them as aerosols (Fig. 9c). In analyzing this scene, we used two
separate versions of the FKM algorithm. Our standard configuration used
*z*_{mid} as one of the classification attributions, while a second, trial
configuration omitted *z*_{mid}. For the two-class FKM, both configurations
successfully identified the high-altitude smoke as aerosol (Fig. 9d and e). But for the three-class FKM, including *z*_{mid} as a classification
attribute introduced uniform misclassification of the lofted smoke as cloud
(Fig. 9f). However, when *z*_{mid} is omitted, the three-class FKM correctly
recognizes the smoke as aerosol (Fig. 9g). This is because including
altitude information can introduce unwanted classification uncertainties
when attempting to distinguish between high-altitude clouds and aerosols,
both of which are located at similar altitudes and have similar optical
properties. Altitude is not a driving factor for classifications and adds
confusion in the memberships defined by the Mahalanobis distance (see Eqs. 7 and 8) in these particular cases. More details are given in
Sect. 5.1. When high-altitude depolarizing aerosols and ice clouds appear at the
same time, either increasing the number of classes to four or omitting
*z*_{mid} as an input will resolve large fractions of the potential
misclassifications from the FKM method.

Figure 10 shows an example of ash from the Kasatochi volcano
(52.2^{∘} N, 175.5^{∘} W), which erupted unexpectedly on 7–8 August 2008 in the central Aleutian Islands. Volcanic aerosols remained
readily visible in the CALIOP images for over 3 months after the eruption
(Prata, et al., 2017). On 5 October 2008, CALIOP observed the “aerosol
plume”
near the tropopause at ∼17:30:18 UTC. COCA V3 classified
those layers with base altitudes above the tropopause as “stratospheric
features” (black regions in Fig. 10b) and misclassified a substantial
portion of the lower tropospheric layers as clouds. Those segments that were
correctly classified as aerosol were frequently assigned low CAD scores. In
contrast, COCA V4 and both versions of the CAD_{FKM} with *z*_{mid} as
inputs show greatly reduced cloud classifications, and the aerosols have
high-confidence CAD scores. Again, when altitude information is not
included, the FKM algorithm produces a better separation of clouds and
aerosols at high altitudes, for the same reasons as in the high-altitude
smoke case.

5 Discussion

Back to toptop
Section 5 compares FKM and COCA using statistical analyses and individual case studies. In this section, we explore the application of various metrics used to evaluate the quality of the FKM and COCA classifications. The questions we address are (a) how much improvement can be made by adding additional measurements as classification inputs (Sect. 5.1); (b) how well the classes are separated (Sect. 5.2); (c) what essential measurements are required for accurately discriminating between clouds and aerosols (Sect. 5.1 and 5.3); and (d) what effects measurement uncertainties (noise) have on the classifications (Sect. 5.4).

Underlying any feature classification task is this essential question: which observations are most important for accurate feature identification? COCA results were substantially improved from V2 to V3 by adding two additional dimensions (latitude and volume depolarization ratio) to the cloud and aerosol PDFs. In general, higher dimension PDFs should improve the classification accuracy so long as the additional dimensions provide some new useful information (i.e., they should be orthogonal, or at least semi-orthogonal, to the data already being used). It is therefore important to quantify how much improvement we can make by adding additional dimensions into the analysis. With the FKM method, it is relatively easy (though perhaps time-consuming) to add or remove one or multiple observational dimensions (i.e., inputs) and reinitiate the training/learning algorithm. (This highly desirable flexibility is, unfortunately, wholly absent in the strictly supervised learning regime incorporated into COCA.) If a dimension is added (or removed) and the new classifications are essentially identical to the old ones, the added (or removed) dimension does not provide significant information in the classification processes. On the other hand, if the CAD values are improved (or degraded) by adding or removing a dimension, this dimension actually contributes dispositive information in the determination of the classification and hence is key to separating clouds from aerosols. By using the FKM method, we can readily determine which parameters are required (and, importantly, which are non-essential) for the resulting classifications to meet predetermined accuracy specifications, either in general or for a particular class (e.g., dust). We can also quantify the improvement (or degradation) that occurs when specific parameters are either added or removed.

We demonstrate these capabilities using individual case study results.
Figure 11 shows series of FKM classifications that omit individual
dimensions from one half orbit of nighttime observations acquired 6 September 2008 (i.e., the same scene shown previously in Fig. 4.) This scene
was chosen as an example specifically because it contains so many
challenging CAD cases (e.g., PSCs, dense water clouds beneath smoke, and
many high-altitude aerosols). Comparisons with COCA V4 and two-class,
four-parameter CAD_{FKM} results are quantified by the confusion matrices
shown in Table 3. From both the figure and the table, we find that the
cloud–aerosol partitioning obtained when any one dimension is omitted is
reasonably similar to the partitioning reported by COCA and by the two-class,
four-parameter FKM algorithm. Both algorithms are most sensitive to the removal
of *χ*^{′} (75.0 % similarity for COCA and 77.4 % similarity
for FKM) and least sensitive to the removal of $\langle {\mathit{\beta}}_{\mathrm{532}}^{\prime}\rangle $ (89.8 % for COCA, 93.1 % for FKM). Note also
from the figure we see that, for the low water clouds covered by a plume of
heavy absorbing smoke, the two-class, four-parameter FKM classifications have low
CAD_{FKM} values. When either *z*_{mid} or $\langle {\mathit{\beta}}_{\mathrm{532}}^{\prime}\rangle $ is removed from the classification parameters, the
CAD_{FKM} values actually improve. Without *χ*^{′} or *δ*_{v}, the CAD_{FKM} values get worse, which indicates that color ratios
and depolarization ratios may play a more important role in separating
aerosols from low water clouds. In this example, the values of *χ*^{′} and $\langle {\mathit{\beta}}_{\mathrm{532}}^{\prime}\rangle $ measured in
the water cloud can bias the resulting CAD_{FKM} values due to the strong
absorption at 532 nm within the overlying smoke layer. $\langle {\mathit{\beta}}_{\mathrm{532}}^{\prime}\rangle $ for these water clouds decreases and gets
closer to the backscatter magnitudes expected from classic aerosols (e.g.,
Fig. 6a), while *χ*^{′} increases far beyond values typical of
classic aerosols. Moreover, when omitting *z*_{mid}, high-altitude aerosols
and ice clouds are more readily and correctly separated, as are low-altitude
aerosols and water clouds. *χ*^{′} or *δ*_{v} are key in
separating high-altitude aerosols and clouds.

In addition to the case study described above, we also analyzed a full month
(January 2008) of CALIOP level 2 data acquired between 60^{∘} S and
60^{∘} N. To better focus on the troposphere, where the vast majority
of detectable atmospheric layers occur, data from the polar regions were
omitted in this test. We assessed the relative importance of various
observational parameters by computing CAD_{FKM} classifications using only
a limited number of inputs (i.e., either one, two, or three of the CALIOP layer
descriptors used in the standard two-class CAD_{FKM} classifications). These
comparisons are summarized in Table 4. The center column of Table 4 shows
the agreement frequencies of these classifications with COCA V4; the right
column shows the agreement frequencies with the two-class, four-parameter FKM
classifications.

When considering those comparisons where only one parameter is removed from the
input data, it is clear that omitting *χ*^{′} has by far the most
deleterious effect. The classifications are relatively insensitive to
omitting any of the other three parameters, though the comparisons are
slightly worse when omitting *z*_{mid} rather than $\langle {\mathit{\beta}}_{\mathrm{532}}^{\prime}\rangle $ or *δ*_{v}. The conclusions to be drawn from
the single-parameter classifications are similar to the three-parameter case,
though perhaps not as stark: using only *χ*^{′} produced slightly
better comparisons with both COCA V4 and the two-class, four-parameter
CAD_{FKM} results than any of the other parameters. Given this
demonstrated sensitivity to *χ*^{′}, it is perhaps not surprising
that of the two-parameter classifications, the combination of *χ*^{′}
and *z*_{mid} proves the most successful. The combination of *χ*^{′} and *δ*_{v} also performed reasonably well relative to both COCA
V4 and the two-class, four-parameter CAD_{FKM}. Unexpectedly, however, the
combination of *χ*^{′} and $\langle {\mathit{\beta}}_{\mathrm{532}}^{\prime}\rangle $ performed very poorly relative to COCA V4, with only
∼67 % of the classifications being identical.

In general, and as expected, the closest matches to the COCA V4 and the
two-class, four-parameter CAD_{FKM} classifications are achieved by the
three-parameter classifications, with the single-parameter classifications
showing the poorest correspondence, and the two-parameter rankings falling
somewhere in between the three-parameter and one-parameter results. However, the
performance of the most successful two-parameter case (the combination of
*χ*^{′} and *z*_{mid}) was largely identical to that of the most
successful three-parameter case (the combination of $\langle {\mathit{\beta}}_{\mathrm{532}}^{\prime}\rangle $, *χ*^{′}, and *z*_{mid}). In fact, relative
to COCA V4, the two-parameter classifications were identical slightly more
often (93.8 % of all cases) than the three-parameter classifications (93.2 %). For the two-class, four-parameter FKM, the corresponding numbers rise
to 95.5 % identity for the best performing two-parameter classifications
and 97.7 % identity for the best performing three-parameter classifications.
Both the COCA and FKM comparisons suggest that the addition of
$\langle {\mathit{\beta}}_{\mathrm{532}}^{\prime}\rangle $ adds little, if any, skill to the
classification task but it contributes to the confidence of the
classifications, as will be shown in Sect. 5.2.

Linear discriminant analysis (Fisher, 1936) is usually performed to
investigate differences among multivariate classes, to validate the
classification quality, and to determine which attributes most efficiently
contribute to the classifications. Here, we introduce Wilks' lambda, which is
the ratio of within-class variance (to evaluate the dispersion within class)
and between-class variance (to examine the differences between the classes).
Considering a data matrix **X** (*n*×*p* matrix, elements *x*_{il}, *i*,
data number $=\mathrm{1},\mathrm{\dots},n;l$, data dimension number $=\mathrm{1},\mathrm{\dots}p$), the FKM
classification returns a membership matrix **M** (*n*×*k* matrix,
elements *m*_{ij}, *i*, data number $=\mathrm{1},\mathrm{\dots},n;j$, class number $=\mathrm{1},\mathrm{\dots},k$) and
centroid matrix **C** (*k*×*p* matrix, elements *c*_{jl}, *j*, class number
$=\mathrm{1},\mathrm{\dots},k;l$, data dimension number $=\mathrm{1},\mathrm{\dots},p$),
where *n* is the number of data samples, *p* is the number of
attributes/dimensions, and *k* is the number of classes. The sum of squares
and products (SSP) within-class covariance matrix **W**_{lm} (*p*×*p* matrix, *l*∕*m*, data dimension number $=\mathrm{1},\mathrm{\dots},p$), also called
the within-class fuzzy scatter matrix (Bezdek, 1981), is given as

$$\begin{array}{ll}{\displaystyle}{\mathbf{W}}_{lm}& {\displaystyle}=\sum _{j=\mathrm{1}}^{k}\sum _{i=\mathrm{1}}^{n}{m}_{ij}^{\mathit{\varphi}}\left({x}_{il}-{c}_{jl}\right)\left({x}_{im}-{c}_{jm}\right),\phantom{\rule{0.125em}{0ex}}\forall \left(l,m\right),\\ \text{(17)}& {\displaystyle}& {\displaystyle}l,m=\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\mathrm{1},\mathrm{2}\mathrm{\dots}p.\end{array}$$

The SSP between-class covariance matrix **B**_{lm} (*p*×*p*
matrix, *l*∕*m*, data dimension number $=\mathrm{1},\mathrm{\dots},p$) is given as

$$\begin{array}{ll}{\displaystyle}{\mathbf{B}}_{lm}& {\displaystyle}=\sum _{j=\mathrm{1}}^{k}\left(\sum _{i=\mathrm{1}}^{n}{m}_{ij}^{\mathit{\varphi}}\right)\left({c}_{jl}-{x}_{l}\right)\left({c}_{jm}-{x}_{m}\right),\phantom{\rule{0.125em}{0ex}}\forall \left(l,m\right),\\ \text{(18)}& {\displaystyle}& {\displaystyle}l,m=\mathrm{1},\mathrm{2}\mathrm{\dots}p.\end{array}$$

The ratio of within class to the total SSP matrix is known as Wilks'
lambda (Eq. 19, Wilks, 1932). Wilks' lambda for multi-dimensional
observations is the determinant of the *p*×*p* matrix, which
represents the geometric volume of this object in *p* dimensions, written as

$$\begin{array}{}\text{(19)}& \mathrm{\Lambda}={\displaystyle \frac{\mathrm{det}\left(\mathbf{W}\right)}{\mathrm{det}\left(\mathbf{W}+\mathbf{B}\right)}}\end{array}$$

(Oh et al., 2005). Here, we use Wilks' lambda (Λ) as a measure of the difference between classes. The value Λ varies from 0 to 1, where 0 suggests that classes differ (within-class SSP is smaller compared to between-class SSP), and 1 suggests that all classes are the same. The magnitude of Wilks' Λ indicates how distinct and well separated the classes are. Smaller values of Wilks' Λ indicate more distinct class separation with minimal between-class overlap; thus, the classifications are more trustworthy and have higher confidence. Wilks' Λ thus provides an additional metric to assess classification algorithm performance, augmenting the classification accuracy indicators shown in Sect. 5.1.

For the January 2008 data, Wilks' Λ for different observational
dimensions is calculated and summarized in Table 5. For 4-D
(*p*=4) observations, Wilks' Λ could be as small as 0.21 for two-class
FKM and even smaller (0.05) for three-class FKM. This means that the classes
generated by the FKM method are well separated, with clusters quite
different from each other, and that the classes in three-class FKM are much
better separated (less overlap in the multi-dimensional observations) than
the classes in two-class FKM. For the two-class FKM, the value of Wilks' Λ
is largest for *z*_{mid}, indicating that, relative to the other individual
parameters, clustering using *z*_{mid} is less efficient at generating
well-separated classes. The large value of Wilks' Λ occurs because
clouds have two distinct altitude centers: one for low water clouds and the
other for high ice clouds. (Mid-level clouds occur too infrequently to form
a third dominant altitude center.) The center altitude of water clouds is
comparable to that of boundary layer aerosols, and thus it is very difficult
to separate these two classes using *z*_{mid} alone. The distinct altitude
centers of ice and water clouds induce large within-class SSP and hence
large values of Wilks' Λ. For single-parameter clustering, Wilks'
Λ from $\langle {\mathit{\beta}}_{\mathrm{532}}^{\prime}\rangle $ is the
smallest, followed by the values for *δ*_{v} and *χ*^{′}.
The value of Wilks' Λ from any combination of observational
dimensions lies between the maximum and minimum values for the
single-parameter clustering. Wilks' Λ values for three-class FKM are much
smaller compared to the two-class FKM values because *z*_{mid} can have an
independent center for each class. For three-class FKM analyses, the largest
single-parameter values of Wilks' Λ are produced by *δ*_{v},
followed by *z*_{mid} and *χ*^{′}. As with two-class FKM, $\langle {\mathit{\beta}}_{\mathrm{532}}^{\prime}\rangle $
yields the smallest value.

In this section, we apply principal component analysis (PCA; Wold et al.,
1987) to the FKM classification results to determine which of the input
parameters account for the greatest variability in the outputs. These
functions, or canonical variants, are therefore calculated from the
eigenvalues and eigenvectors of matrix **W**_{f}∕**B**_{f} (the ratio of
within-class variance and between-class variance). The first function
(PCA-1) maximizes the differences between the classes and represents the
dominant contribution to the classifications. Successive functions (PCA-2)
will be orthogonal to, or independent of, the other functions, and hence
their contributions to the discrimination between classes will not overlap.
We also project the inputs' variable vectors along the principal component
axes. Using this method helps to better understand how independent the input
parameters are and how they individually contribute to the classifications.

The scatter plots of PCA-1 and PCA-2 for two-class and three-class FKM are
shown in Fig. 12. The projections of vector lengths on PCA-1 and PCA-2 of
different measurements (i.e., $\langle {\mathit{\beta}}_{\mathrm{532}}^{\prime}\rangle $, *δ*_{v}, *χ*^{′}, and
*z*_{mid}) indicate how much each individual dimension contributes to the
classifications. Longer projections mean stronger contributions. From the
figure, we clearly see that water clouds, ice clouds, and aerosols are quite
different (i.e., their cluster centers are located in different positions).
Different colors represent different classes, and darker colors indicate
higher sample densities. Class centers, marked with red crosses, are located
where the class sample density is highest, with higher densities shown by
darker colors. We reorient PCA-2 to keep the C1–C2 line approximately
diagonal and thus better assess the relationship between PCA-1 and PCA-2.
(In reality, the contribution of PCA-1 is always larger than PCA-2, while the
diagonal line shows PCA-1 contribution is equal to PCA-2.) From both panels,
we see that class 1 (cloud) of the two-class FKM breaks into two classes (ice cloud
and water cloud) when applying the three-class FKM. The denser samples (centers) of
water cloud, ice cloud, and aerosol are quite separate from each other, and
the overlap zone has fewer samples. We can also see that *χ*^{′}
and *δ*_{v} contribute the most to PCA-1 (longer projections on the
axis of PCA-1 in both subpanels), while $\langle {\mathit{\beta}}_{\mathrm{532}}^{\prime}\rangle $ and *z*_{mid} contribute more to
PCA-2. Hence, *χ*^{′} and *δ*_{v} are the driving
components for the cloud–aerosol separation. From Fig. 12b, we could also
argue that $\langle {\mathit{\beta}}_{\mathrm{532}}^{\prime}\rangle $ and *δ*_{v} are the driving factors in classifying water and ice clouds
(projections of the vectors on C2–C3, namely the combined projection of
PCA-1 and PCA-2, are longer), while *χ*^{′} and *z*_{mid} also
contribute to the classification. *z*_{mid} and, to a greater extent, *χ*^{′} and *δ*_{v} are the driving factors that allow aerosols
to be separated from ice clouds (projections of the vectors on C1–C2 are
longer), whereas and *χ*^{′} are the driving factors that separate
water clouds from aerosols (projections of the vectors on C1–C3 are longer).
Comparing contributions of individual measurements to different classes,
*z*_{mid} is most useful in helping discriminate aerosols and ice clouds,
while simultaneously being the least useful in separating aerosols and water
clouds. $\langle {\mathit{\beta}}_{\mathrm{532}}^{\prime}\rangle $ is the most useful
parameter in distinguishing aerosols form water clouds and water clouds from
ice clouds, and the least useful in differentiating between aerosols and ice
clouds. *δ*_{v} is most useful in distinguishing between water
clouds and ice clouds and between aerosols and ice clouds, and the least
useful in separating aerosols from water clouds. These observations agree
very well with earlier findings in Fig. 6 and Tables 2, 3, and 4.

By using PCA, we can determine which parameters are most influential in
arriving at different cluster memberships. Additionally, because all CALIOP
measurements are contaminated to some degree by noise, we also want to see
if/how noise in the individual parameters affects classification accuracy.
These results can also guide us in understanding how the classification
accuracy changes as the CALIOP laser energies deteriorate over the lifetime
of the mission. In this section, we assess the impacts of instrument noise
and measurement uncertainties on the FKM classifications. The observations
from a nighttime granule acquired 6 September 2008 beginning at
∼01:35:29 UTC are used to investigate how noise in the lidar
measurements affects the accuracy of the clustering results and what, if
any, errors are introduced into the cloud and aerosol classifications for
this particular case. To simulate the measurement uncertainties, two
different methods are used. The first drew pseudo-random variables from
Gaussian distributions having means equal to the various measured values and
standard deviations between 10 % and 200 % of the means. As
illustrated in Fig. 13, using this method allows us to quantify the
effects of varying measurement errors on the FKM classification algorithm
results. A sequence of Monte Carlo tests was constructed in which one of the
four classification variables was randomly perturbed (i.e., drawn from the
aforementioned Gaussian distributions), while the other three remained
unchanged. For each of the four tests, 100 realizations of simulated input
were created. To estimate the propagation of measurement uncertainties, we
calculated the shifts in classification, confusion indices (CIs; see Sect. 3.2), and the changes in cluster centers between new clusters with added
noise and the original clusters derived using unperturbed inputs. The shifts
in cluster centers are the mean distances between the centers of the new
clusters (*C*_{n}, obtained from perturbed dataset) and the old ones
(*C*_{o}, obtained from error-free dataset) for both clouds and aerosols,
calculated using Eq. (20) (Omar et al., 2005). These distances are normalized
by the standard deviation of the distributions (*C*_{std}) of individual
record distances from unperturbed center as

$$\begin{array}{}\text{(20)}& \mathit{\delta}{d}_{\mathit{\epsilon}}={\displaystyle \frac{\left|{\mathit{C}}_{\mathrm{n}}-{\mathit{C}}_{\mathrm{o}}\right|}{{C}_{\mathrm{std}}}},\end{array}$$

where $\left|x\right|$ represents the L1 norm of *x*. Figure 13 plots (a)
the shifts in cluster centers for each class, (b) the fraction of correct
classifications, and (c) the revised confusion indices as a function of
relative uncertainties ranging from 10 % to 200 %. From Fig. 13a, we
see that shifts in cluster centers between perturbed and unperturbed data
are very small when the uncertainties are small. The largest shift comes
from color ratio perturbations and the smallest shift comes from backscatter
perturbations. Perturbations on class 2 (aerosol) are more important
compared to class 1 (cloud). Figure 13b and c show that when the
uncertainties in the measurements are small (i.e., less than 10 %), the
errors in the classifications are also small (e.g., less than 2 % in
Fig. 13b) with less overlaps between classes (e.g., small values of CI
from 0.3 to 0.305 seen in Fig. 13c). When the uncertainties increase, the
classification accuracies slightly decrease and the shifts in cluster center
and CI slightly increase. The rates of change in the accuracy and confusion
index are rapid at first (i.e., between relative uncertainties between 10 % and 100 %) but tend to stabilize for larger uncertainties. Large
measurement uncertainties (i.e., 200 %) in color ratio can introduce
biases of 20 % in the classification results, with CI values less than
0.335. This suggests that uncertainties in the measurements can cause
misclassification but that most of the classifications (∼80 %) are still robust. This is because cloud and aerosol properties are
largely distinct and the misclassifications that do occur may come from
features such as the few very thin clouds and dense aerosols in the
transitional zone in Fig. 6.

Our first error propagation test used arbitrarily assigned relative
uncertainties between 10 % and 200 % of the parameter mean values. In
our second test, we used the measured uncertainties reported in the CALIOP
layer products to construct the Gaussian distributions from which
pseudo-random variables were generated. By using this method, we can assess
the actual impacts on the classifications due to noise in the CALIPSO
measurements. To isolate the influence of the individual inputs, three test
cases were constructed in which only one parameter was varied in each case.
Figure 14 shows the results. Figure 14a shows the unperturbed results, while
Fig. 14b–d show CAD_{FKM} scores averaged over 100 perturbations of
the test parameter. Figure 14b shows the results when the attenuated
backscatter intensities are varied, Fig. 14c shows the results when the
depolarization ratios are varied, and Fig. 14d shows the results when the
color ratios are varied.

From Fig. 14, we find that the averaged CAD_{FKM} scores from the
perturbed datasets do not differ markedly from the CAD_{FKM} scores in the
unperturbed dataset. In more than 88 % of the cases, clouds are still
classified as clouds and aerosols are still classified as aerosols. When
examining perturbations to backscatter intensity alone (Fig. 14b), we find
that the perturbed and unperturbed classification results are identical more
than 98 % of the time. However, the CAD_{FKM} differences arising from
perturbations to depolarization ratio and color ratio (Fig. 14c and d,
respectively) can be much larger. This finding is consistent with results
shown earlier in Fig. 13.

Most often, the perturbed measurements only induce CAD_{FKM} changes for
features that were originally classified with low confidence and for those
challenging features such as water clouds beneath smoke, high-altitude
aerosols, and PSCs, whose input parameters frequently lie in the transition
zone between clouds and aerosols. Water clouds beneath thick smoke layers
are an especially difficult case, as the uncertainties introduced by the
absorption of smoke at 532 nm can significantly reduce the confidence of the
water cloud classification. Looking at Fig. 14, together with Figs. 2,
11, and 12, we find that this is a reasonable and even expected result. From
Figs. 11 and 14, we know that the most effective measurements for
separating water clouds and aerosols are color ratio and backscatter
intensity. But relative to measurements of water clouds in otherwise clear
skies, the color ratios for water clouds lying under absorbing smoke layers
have large positive biases, while the backscatter intensities have large
negative biases, and these biases will produce low-confidence CAD scores,
both for the FKM method and the V4 operational method (Liu et al., 2019). A
somewhat similar scenario can occur in the classification of high-altitude
aerosols, where high biases (i.e., measurement errors) in *δ*_{v}
and *χ*^{′} can lead to the misclassification of aerosols as ice
clouds.

6 Conclusions

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In this paper, we use the fuzzy *k*-means (FKM) clustering algorithm to
evaluate the classifications reported by the cloud–aerosol discrimination
(CAD) algorithm used in the standard processing of the Cloud-Aerosol Lidar
with Orthogonal Polarization (CALIOP) measurements. Being able to accurately
separate clouds from aerosols is an essential task in the analysis of the
elastic backscatter lidar measurements being continuously acquired by
Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observation (CALIPSO)
mission. When coupled to a well-validated CAD algorithm, the data products
delivered by CALIOP can be used to reliably map the vertical distributions
of clouds and aerosols on global and regional scales throughout the full 12 years of the CALIPSO mission.

The comparison between two different classification techniques helps us assess the performance of the operational CAD algorithm from the same scenes. The CALIOP operational CAD algorithm (COCA) is a supervised learning technique in which classification decisions are tuned to match externally provided expert human judgements. FKM is an unsupervised learning scheme, which assigns class memberships based on similarities discovered in the inherent characteristics of the input data. While the two algorithms both rely on the same underlying lidar measurements, the underlying mathematical formulations are entirely different, as is the framework for expressing class membership values. These differences allow us to explore the classification uncertainties due to the algorithms. The flexibility of the FKM technique also allows us to investigate the relative importance of various inputs in deriving the final classifications and to explore classification misclassification arising from current lidar measurement techniques. Establishing these performance metrics should enable the development of enhanced classification schemes for use with future space-based lidars.

The key finding of this study is that the feature classifications assigned by COCA are very closely replicated by the FKM method. Having a totally unsupervised learning algorithm “discover” the same patterns in the data that are reported by COCA strongly suggests that the COCA classifications represent genuine data-driven differences in the CALIOP observations and are thus largely free from artifacts that might be imposed by human misinterpretations when constructing the CAD probability density functions (PDFs). The classifications obtained from our independently derived FKM analyses compare well with the classifications determined by COCA and reported in the CALIOP V4 data products. Using a 1-month test set, the two-class and three-class FKM classifications agreed with the V3 and V4 operational data products over 93 % of the time, and the three-class FKM results agreed with the COCA results in 94 %–95 % of all cases. This strong agreement between two independent methods provides convincing evidence that the V4 operational CAD algorithm is delivering robust and accurate classifications.

Those instances where the two methods fail to agree (5 %–6 %
of all cases) are typically highly ambiguous scenes in which the observables
lie in the overlap regions between the peaks of the cloud and aerosol PDFs.
In particular, in scenes containing Taklamakan dust (or lofted Asian dust in
general), high-altitude smoke plumes, and/or volcanic ash, both the V4
operational CAD and the FKM algorithm struggle to make accurate
classifications. The Taklamakan dust cases provide an instructive example
that illustrates the classification conundrum. Over the Taklamakan Desert, lofted
dust layers and cirrus clouds occur in similar temperature regimes and
frequently have similar backscatter intensities ($\langle {\mathit{\beta}}_{\mathrm{532}}^{\prime}\rangle $) and depolarization ratios (*δ*_{v}). The
most critical criterion for distinguishing clouds from aerosols is color
ratio (*χ*^{′}), and the characteristic color ratios of dust and
cirrus are reasonably distinct (∼0.75 versus ∼1.01). However, the natural variability within each feature type is quite
broad (e.g., ±0.25 for cirrus), and the measurements are very noisy,
especially during daytime.

To characterize the CAD improvements made in the most recent CALIOP data release, we used the FKM method to explore the capabilities of both the V3 and V4 operational CAD algorithms. As expected, the V4 operational algorithm was more effective than the V3 version. The primary differences are found by examining the results obtained for specific feature classes. The FKM classifications agree well with both the V3 CAD results in most cirrus fringe and dense aerosol cases and agree well with V4 CAD results for lofted Asian dust, high-altitude smoke, and volcanic ash. FKM classifications of stratospheric features and polar region features had the largest uncertainties. More studies are needed to better understand why these specific types of features are proving so resistant to confident classification, irrespective of the algorithmic approach applied.

Our investigation of error propagation in the FKM shows that while measurement uncertainties on the order of the CALIPSO measured noise will introduce biases into the cloud and aerosol classifications, more than 80 % of the classifications stay unchanged. For the rest of classifications (which are low-confidence clouds or aerosols), as the uncertainties increase, the classification confidence decreases, as indicated by higher confusion indexes, and the classification accuracies decrease as well. The dependence and the number of measurements can also impact the classification efficiency. Key parameter analysis shows that higher classification accuracies are achieved by increasing the number of independent observational parameters used in the analyses. Two-class FKM classifications using only a single input yield the same results as the operational CAD in only ∼60 % of all classifications, a rate only marginally better than would be expected from random choice. While using three parameters achieved an agreement between the FKM and the V4 operational CAD in the neighborhood of ∼80 %, raising the agreement to ∼95 % required four parameters. When only three inputs were used, removing the color ratio from the FKM caused the largest classification disparities between the two methods.

Certain parameters are especially significant for the classification of
particular feature types, and thus optimizing the number of successful
classifications across all features requires the inclusion of all
measurements that effectively contributed to any species-specific
classification. Principal component analysis and key parameter analysis
together show that the most important dimensions for distinguishing between
clouds and aerosols are *δ*_{v} and *χ*^{′}; that
$\langle {\mathit{\beta}}_{\mathrm{532}}^{\prime}\rangle $ and *δ*_{v} are the driving
factors in classifying water and ice clouds; and that altitude (*z*_{mid}),
*χ*^{′}, and *δ*_{v} are the key inputs that allow aerosols
to be separated from ice clouds, while $\langle {\mathit{\beta}}_{\mathrm{532}}^{\prime}\rangle $ is the critical factor for separating
aerosols and water clouds. Moreover, from fuzzy linear discriminant analysis,
we found the values of Wilks' lambda are close to 0, confirming that the FKM
classification technique reliably separates clouds from aerosols.

The flexibility of the FKM method offers opportunities to explore the effectiveness of future classification schemes that potentially incorporate measurements from multiple sensors, perhaps even from multiple satellites. While the input data used by our implementation of the FKM technique are essentially synthetic to those required by the CALIOP V4 operational algorithm, the two decision-making frameworks are independently derived and rely on very different mathematics (i.e., probabilities versus fuzzy logic). The very close similarity between the results produced by the two independent approaches argues strongly that the V4 operational classifications are essentially correct at the 94 % level.

Data availability

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Data availability.

The analyses in this studied relied on the V4.10 CALIPSO level 2 vertical feature mask product (Vaughan et al., 2018; NASA Langley Research Center Atmospheric Science Data Center; https://doi.org/10.5067/CALIOP/CALIPSO/LID_L2_VFM-Standard-V4-10) and the V4.10 CALIPSO level 2 5 km merged layer product (Vaughan et al., 2018; NASA Langley Research Center Atmospheric Science Data Center; https://doi.org/10.5067/CALIOP/CALIPSO/LID_L2_05kmMLay-Standard-V4-10). The CALIPSO level 2 data products are also available from the AERIS/ICARE Data and Services Center (http://www.icare.univ-lille1.fr, AERIS/ICARE; last access: 22 May 2018).

Code availability

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Code availability.

Source code for the fuzzy *k*-means algorithm used in this study was
downloaded from the website of the University of Sydney:
https://sydney.edu.au/agriculture/pal/software/fuzme.shtml (Minasny and McBratney, 2002).

Competing interests

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Competing interests.

The authors declare that they have no conflicts of interest. Author Charles Trepte is a guest co-editor for the “CALIPSO version 4 algorithms and data products” special issue in Atmospheric Measurement Techniques but did not participate in any aspects of the editorial review of this paper.

Special issue statement

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Special issue statement.

This article is part of the special issue “CALIPSO version 4 algorithms and data products”. It is not associated with a conference.

Review statement

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Review statement.

This paper was edited by James Campbell and reviewed by three anonymous referees.

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Special issue

Short summary

We use a fuzzy *k*-means (FKM) classifier to assess the ability of the CALIPSO cloud–aerosol discrimination (CAD) algorithm to correctly distinguish between clouds and aerosols detected in the CALIPSO lidar backscatter signals. FKM is an unsupervised learning algorithm, so the classifications it derives are wholly independent from those reported by the CAD scheme. For a full month of measurements, the two techniques agree in ~ 95 % of all cases, providing strong evidence for CAD correctness.

We use a fuzzy *k*-means (FKM) classifier to assess the ability of the CALIPSO cloud–aerosol...

Atmospheric Measurement Techniques

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