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  • View in gallery

    Plane position indicator (PPI) of (a) Zh (dBZ), (b) Zdr (dB), (c) Kdp (° km−1), (d) , and (e) clustering result at the 3.1° elevation angle at 1400 UTC 19 Nov 2008.

  • View in gallery

    Flowchart of the system for hydrometeor classification. Part I describes the derivation of PDFs for different hydrometeor types, and Part II proposes two classification algorithms using the PDFs.

  • View in gallery

    Flowchart of the prototype generation unit.

  • View in gallery

    Scatterplots of the (a) well-behaved cluster and (b) mixed-habit cluster.

  • View in gallery

    The Zh histogram of the (a) mixed-habit cluster, (b) detected peaks and valleys of the histogram, and (c) clustering result.

  • View in gallery

    (left) A cluster and (right) the derived prototype for moderate rain, observed at 0624 UTC 16 Nov 2008. The projection of five-dimensional distribution is displayed in a panel of (a),(b) ZhZdr; (c),(d) ZhKdp; (e),(f) Zh; and (g),(h) ZhTenv. In the first column, the areas with data points less than 10% of the total points are screened out, and the color bars are the number of data points. In the second column, the black marks are the means of the prototypes, and the color bars represent the density of the prototype.

  • View in gallery

    Original prototypes on (a) ZhZdr fields and (b) reduced prototypes. The colored marks are the means of the prototypes.

  • View in gallery

    Comparison of the (a)–(d) prototype means and (e)–(h) PDFs between rain categories [light rain (green), moderate rain (blue), and heavy rain (red)].

  • View in gallery

    PDFs for light rain. The projection of the five-dimensional distribution is displayed: (a) ZhZdr, (b) ZhKdp, (c) Zh–ρhv, and (d) ZhTenv. The black dots represent the prototype means, and the thresholds plotted follow the ones in Straka et al. (2000). The key shows Rl: light rain, Rm: moderate rain, Rh: heavy rain, and R: rain.

  • View in gallery

    As in Fig. 9, but for moderate rain.

  • View in gallery

    As in Fig. 9, but for heavy rain.

  • View in gallery

    As in Fig. 9, but for hail, where G/Hs: graupel/small hail, R/H: rain/hail mixture, and H: hail.

  • View in gallery

    As in Fig. 12, but for the rain/hail mixture.

  • View in gallery

    As in Fig. 12, but for graupel/small hail.

  • View in gallery

    Comparison of the (a)–(d) prototype means and (e)–(h) PDFs between graupel/small hail (green), rain/hail mixture (blue), and hail (red).

  • View in gallery

    Comparison of the (a)–(d) prototype means and (e)–(h) PDFs between ice crystals (green), ice aggregates (blue), and melting ice (red).

  • View in gallery

    As in Fig. 9, but for ice aggregates, where IC: small ice crystals, IA: ice aggregates, MI: melting ice, ICh: horizontally oriented crystals, and ICυ: vertically oriented crystals.

  • View in gallery

    As in Fig. 17, but for melting ice.

  • View in gallery

    As in Fig. 17, but for ice crystals.

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A Cluster-Based Method for Hydrometeor Classification Using Polarimetric Variables. Part I: Interpretation and Analysis

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  • 1 LACS, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing, China
  • | 2 Centre for Australian Weather and Climate Research, Bureau of Meteorology, Melbourne, Victoria, Australia
  • | 3 University of Melbourne, Melbourne, Victoria, Australia
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Abstract

Hydrometeor classification methods using polarimetric radar variables rely on probability density functions (PDFs) or membership functions derived empirically or by using electromagnetic scattering calculations. This paper describes an objective approach based on cluster analysis to deriving the PDFs. An iterative procedure with K-means clustering and expectation–maximization clustering based on Gaussian mixture models is developed to generate a series of prototypes for each hydrometeor type from several radar scans. The prototypes are then grouped together to produce a PDF for each hydrometeor type, which is modeled as a Gaussian mixture. The cluster-based method is applied to polarimetric radar data collected with the CP-2 S-band radar near Brisbane, Queensland, Australia. The results are illustrated and compared with theoretical classification boundaries in the literature. Some notable differences are found. Automated hydrometeor classification algorithms can be built using the PDFs of polarimetric variables associated with each hydrometeor type presented in this paper.

Corresponding author address: Guang Wen, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China. E-mail: g.wen@student.unimelb.edu.au

Abstract

Hydrometeor classification methods using polarimetric radar variables rely on probability density functions (PDFs) or membership functions derived empirically or by using electromagnetic scattering calculations. This paper describes an objective approach based on cluster analysis to deriving the PDFs. An iterative procedure with K-means clustering and expectation–maximization clustering based on Gaussian mixture models is developed to generate a series of prototypes for each hydrometeor type from several radar scans. The prototypes are then grouped together to produce a PDF for each hydrometeor type, which is modeled as a Gaussian mixture. The cluster-based method is applied to polarimetric radar data collected with the CP-2 S-band radar near Brisbane, Queensland, Australia. The results are illustrated and compared with theoretical classification boundaries in the literature. Some notable differences are found. Automated hydrometeor classification algorithms can be built using the PDFs of polarimetric variables associated with each hydrometeor type presented in this paper.

Corresponding author address: Guang Wen, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China. E-mail: g.wen@student.unimelb.edu.au

1. Introduction

The aim of hydrometeor classification with polarimetric variables is to assign precipitating echoes, or specific regions of precipitating echoes, to one of several hydrometeor categories based on backscattering models of electromagnetic waves. This is because polarimetric variables are sensitive to habits, shapes, sizes, and other microphysical properties of hydrometeors. The accurate classification of hydrometeor type is crucial for weather radar applications. The occurrence of ice-phase and mixed-phase scatterers affects the accuracy of quantitative precipitation estimation (Giangrande and Ryzhkov 2008; Cifelli et al. 2011). Discrimination between hail and rain is also important in determining whether to issue severe weather warnings (Keenan 2003). Hydrometeor classifications can also be used to cross validate observations from spaceborne radars and ground-based polarimetric radars (Chandrasekar et al. 2008) and to study the microphysical structure of convective thunderstorms (Vivekanandan et al. 1999).

Since the mid-1970s, research has demonstrated the potential for the discrimination of various hydrometeor scatterers with polarimetric variables, for example, Seliga and Bringi (1976), Hall et al. (1984), Seliga et al. (1986), Aydin et al. (1986), Balakrishnan and Zrnić (1990), Straka and Zrnić (1993), and Doviak and Zrnić (2006). Methods with hard decision boundaries, such as the decision tree method, were developed based on the studies in the above-mentioned papers. However, the ranges of polarimetric variables for different hydrometeor types are not always exclusive to set hard thresholds. To improve these hydrometeor classifications, Straka (1996), Vivekanandan et al. (1999), Liu and Chandrasekar (2000), Zrnić et al. (2001), Schuur et al. (2003), and other researchers introduced a semiempirical rule-based fuzzy logic classification methodology to allow the boundaries to cross each other, yielding a “soft” classification. Straka et al. (2000) synthesized the existing studies to produce relationships between polarimetric variables and microphysical properties of hydrometeors for use in hydrometeor classification techniques. Lim et al. (2005) and Park et al. (2009) added new features to the fuzzy logic–based algorithms. Although the hydrometeor classification techniques were originally developed for S-band radars, they have been adapted for other radar frequencies, such as C band (e.g., Keenan 2003; Marzano et al. 2007) and X band (e.g., Dolan and Rutledge 2009). A variety of field experiments, including the well-known Joint Polarization Experiment (Ryzhkov et al. 2005), demonstrated the performance of the fuzzy logic algorithms. Chandrasekar et al. (2013) comprehensively reviewed the advances of the hydrometeor classification algorithms. Marzano et al. (2008) and Li (2011) also developed probabilistic methods based on -matrix scattering calculations.

While these studies undoubtedly demonstrate that polarimetric variables contain quantitative information about hydrometeor types, there are arguably still some gaps in our knowledge of the polarimetric signatures of hydrometeors. Therefore, there is room for improvement of the hydrometeor classification methods regarding the following aspects:

  1. There is a lack of in situ measurements and ground truth for some hydrometeor types, especially ice hydrometeors. As a consequence, the true probability density functions (PDFs) associated with these hydrometeors are largely unknown.

  2. The shapes of the PDFs are preassumed, for example, modified beta functions and trapezoidal functions. The variables are generally considered individually in these PDFs, but the variables may not be completely independent.

  3. The specific characteristics of radar signal quality are not naturally included in the design of hydrometeor classification methods. Noise in the radar-measured quantities presumably has an impact on the radar-derived PDFs of polarimetric variables describing each hydrometeor type. This convolution of the true PDFs with the radar noise characteristics is not completely captured in the existing hydrometeor classification methods.

In this series of two papers (G. Wen et al. 2015, unpublished manuscript, hereinafter Part II) we describe a new method to overcome some of these issues. A clustering method is used in the present paper (Part I) in order to derive the PDFs of the polarimetric variables for different hydrometeor types. A number of clusters are first derived from a series of volumetric scans, and then validated and labeled to build prototypes, which are defined as the distributions of these clusters. Each prototype is represented by a Gaussian distribution with a weight. The PDFs for each hydrometeor type can be regarded as a Gaussian mixture built from these prototypes. The derived PDFs form the observational basis from which new hydrometeor classification algorithms can be developed. In the second paper, Part II, we propose two such algorithms using these PDFs, namely, a prototype-level maximum prototype likelihood classifier (MPLC) and a distribution-level Bayesian classifier. It is to be noted that the methodology presented in these papers can be applied to any weather radar frequency.

The expected advantages of a clustering technique may be summarized as follows:

  1. The generation of prototypes and associated PDFs does not rely on preconceived ideas about the shape of the PDF. Each cluster is identified by the technique because the set of polarimetric variables describing it is objectively different from the others. As a result, the identification of different hydrometeor types comes directly from the clustering technique rather than ad hoc assumptions.

  2. It is an incremental method, in the sense that more prototypes can be added as more radar data are processed. In other words, the accuracy of the PDFs can be improved by generating more prototypes as more data become available.

  3. Differences in radar noise characteristics are naturally included in the derivation of the PDFs for different radar types or locations.

In section 2, the definition of polarimetric radar variables and their potential for hydrometeor classification are discussed. In section 3, the clustering method developed to derive the PDFs for different hydrometeor types is presented and discussed. The PDFs derived from S-band radar data collected by the CP-2 radar in Brisbane, Australia, are then analyzed and compared to theoretical classification boundaries in section 4. Finally, we summarize the findings of this paper in section 5.

2. Polarimetric variables

Polarimetric variables are derived from the covariance matrix of the polarized radar return signals. These variables contain considerable information about the hydrometeors within the illuminated volume, such as habits, shape, size, fall behavior, and other microphysical properties (Bringi and Chandrasekar 2001). Deriving the PDFs of the polarimetric variables for each hydrometeor type from radar observations or electromagnetic scattering calculations is an important task upon which hydrometeor classification techniques rely.

Four polarimetric variables are considered in this study, namely, the reflectivity factor (Zh), the differential reflectivity (Zdr), the correlation coefficient at zero lag (), and the specific differential phase (Kdp). The first three power-based variables are related to backscattering amplitudes from hydrometeors within a radar resolution volume. Term Kdp is the range derivative of the differential phase shift (), which is a propagation variable that reflects the integrated properties of the medium along a single radial (Doviak and Zrnić 2006). These four polarimetric variables are briefly described below. Table 1 summarizes typical relative values of polarimetric variables for different hydrometeor types. Figure 1 illustrates typical signatures associated with different hydrometeor types.

Table 1.

Relative intensity of polarimetric variables and environmental temperature expected for different hydrometeor types.

Table 1.
Fig. 1.
Fig. 1.

Plane position indicator (PPI) of (a) Zh (dBZ), (b) Zdr (dB), (c) Kdp (° km−1), (d) , and (e) clustering result at the 3.1° elevation angle at 1400 UTC 19 Nov 2008.

Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-13-00178.1

a. Reflectivity factor

The reflectivity factor of the horizontal polarized signal is defined as (Bringi and Chandrasekar 2001; Doviak and Zrnić 2006)
e1
where D is the equivalent diameter, N(D) is the drop size distribution in a resolution volume, represents the horizontal radar cross section, K is the dielectric complex factor, is the radar wavelength, and shh is the amplitude of the horizontally transmitted and horizontally received backscattering wave. In the Rayleigh scattering regime, the cross-section is proportional to the sixth-order moment of the size distribution, so Zh is largely dominated by the largest particles within a volume. Equation (1) also shows that Zh is affected by the dielectric constant K, which produces a difference between ice-phased and liquid-phased hydrometeors of the same particle size (Aydin et al. 1986; Ryzhkov and Zrnić 1998; Ryzhkov et al. 1998). Term Zh is also very useful to identify melting ice because a liquid-coated (melting) ice particle produces larger Zh than a pure ice particle of the same diameter (the so-called brightband effect). An example of polarimetric radar observations is illustrated in Fig. 1. The circle of high Zh at a radar range of about 60 km corresponds to melting ice. The explanation of the Zh increase is described in more detail in Williams et al. (2007) and Battan (1973). From this circle of enhanced Zh toward the radar location, larger raindrops can be identified by the larger Zh, while smaller Zh outside the circle may be produced by ice particles (ice aggregates or pristine ice crystals). In addition, hail usually has a large size and thus produces a signature of high Zh.

b. Differential reflectivity

The differential reflectivity is defined as
e2
where Zυ is the reflectivity factor at the vertical polarization and svv is the amplitude of the vertically received return backscattered from a vertically transmitted wave. Term Zdr is influenced by the axis ratio, density, size, and canting effects of hydrometeors (Seliga and Bringi 1976). Oblate spheroids such as raindrops produce positive Zdr; in contrast, prolate spheroids produce negative Zdr. Hydrometeors with near-unity apparent axis ratio have Zdr close to zero. In Fig. 1, the Zdr of larger raindrops (characterized by larger Zh) is systematically larger than the Zdr of smaller raindrops (characterized by smaller Zh), which is due to the well-known fact that raindrops become more oblate as they grow. The regions of negative Zdr above the freezing level may indicate ice crystals with a preferential vertical orientation (Illingworth et al. 1987; Brandes et al. 1995). In addition, the combination of Zh and Zdr effectively distinguishes hail from heavy rain, because heavy rain produces large Zh and large Zdr, while hail is characterized by large Zh but relatively small Zdr. The Zh and Zdr resulting from a mixture of rain and hail will be dominated by the hail, since these measurements are weighted by the largest particles (hail) in the size distribution (Bringi and Chandrasekar 2001; Doviak and Zrnić 2006).

c. Correlation coefficient at zero lag

The correlation coefficient at zero lag measures the degree of decorrelation between horizontally and vertically polarized signals, which is defined as
e3
where the asterisk (*) denotes the complex conjugate.

The decorrelation results from the variability in the horizontal and vertical orientations of particles in a radar resolution volume. Low is expected for a mixture of different thermodynamic phases or habits within the radar sampling volume, for example, melting ice and rain/hail mixture. Conversely, is usually larger than 0.95 when a single hydrometeor type is dominant in a particular radar range gate. In addition, is useful for quality control of radar measurements, because nonmeteorological scatterers produce much lower values than weather echoes. In Fig. 1, the circle of low is the signature of melting ice. The inside and outside the circle may be produced by rain and ice aggregates, respectively. Note the low values of at far ranges near echo edges may result from low signal-to-noise ratio.

d. Specific differential phase

The specific differential phase is the range derivative of differential propagation phase−between horizontally and vertically polarized waves. It is formulated as
e4
where fhh and fvv are the horizontally and vertically polarized forward scattering amplitudes, respectively. The Kdp is a measure of the gradient of the local phase shifts due to anisotropic scatterers, because they produce different phase shifts for horizontally and vertically polarized waves. The Kdp is almost proportional to rainfall rate when rain is not mixed with ice. Thus, it can be used to discriminate various intensities of rain. In Fig. 1, high Kdp within the circle of high Zh is indicative of heavy rain with large, oblate raindrops. Low Kdp outside the circle corresponds to the ice-phased particles.

In addition, the environmental temperature (Tenv) obtained from a surrounding operational radiosonde or from outputs of a numerical weather prediction model is used. It is important to note that the temperature profile used here is a single profile collected some distance from the radar, and some hours before or after radar observations. The temperature information is a constraint on the hydrometeor classification to separate liquid-phase and ice-phase hydrometeors. Owing to the very intermittent availability in time and space of temperature, this information should be used with caution and some flexibility needs to be allowed in retrieval techniques to allow for ice particles above freezing level and liquid particles below freezing level.

3. Cluster-based method

The objective of this paper is to develop a clustering technique to derive the PDFs of polarimetric variables and temperature associated with different hydrometeor types. To apply this technique to any radar, the first step is to build a training radar dataset that includes a sufficient number of radar volumetric scans to make sure that all possible types of hydrometeors are included. For instance, hail cases should definitely be included in hail-prone areas. To demonstrate the potential of this clustering technique, we have included 25 cases; however, ideally more data should be included in order to capture the possible ranges of large-scale conditions and storm types typical of the region where the radar is installed. Once the cluster analysis has been applied to a sufficient number of cases, the PDFs will not need to be altered for a given radar and location. If the radar characteristics change dramatically or the radar is moved to another location, then the clustering technique would need to be applied to a new radar dataset in order to update the PDFs. Hydrometeor classification techniques using these PDFs can then be developed and used operationally or for research purposes (see Part II of this study for two examples of use of the PDFs derived in this paper for this specific radar and location).

Figure 2 illustrates the framework of the cluster-based classification method. Four polarimetric variables—Zh (dBZ), Zdr (dB), Kdp (° km−1), and —and the environmental temperature Tenv (°C) are used as inputs to the method. Term Kdp is estimated from the measured differential phase shift () over 14 successive range gates (2 km) by using a linear regression-based method described in Bringi and Chandrasekar (2001). The polarimetric data used here were postprocessed by a quality control procedure proposed by Bringi and Thurai (2008). It eliminates the radar echoes from nonmeteorological scatterers, including anomalous propagation, ground clutter, birds, and insects. The steps for the quality control can be summarized as follows:

  1. A threshold of 10° for the standard deviation of is applied to mask out nonmeteorological echoes (a 10-gate running-mean estimate of the standard deviation, which corresponds to 1.5 km with the 150-m CP-2 range sampling).

  2. Term is corrected for noise using the signal-to-noise ratio (SNR; Schuur et al. 2003) as , where .

  3. A boxcar smoothing over two azimuths and two range gates on either side (i.e., over five gates) is used for Zh, Zdr, and to reduce noise in the raw data.

Fig. 2.
Fig. 2.

Flowchart of the system for hydrometeor classification. Part I describes the derivation of PDFs for different hydrometeor types, and Part II proposes two classification algorithms using the PDFs.

Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-13-00178.1

Future work will aim to develop classification schemes for nonmeteorological scatterers using the cluster-based type of method described here. It would utilize variables not used in this paper, such as and the textures of the polarimetric variables.

The overall cluster-based method for hydrometeor classification consists of two major components: the prototype generation unit and the hydrometeor classification unit. This paper discusses the methodology for generating a number of prototypes and deriving PDFs from these prototypes. The algorithms for hydrometeor classification based on the clustering-derived prototypes and PDFs is described in Part II.

Figure 3 shows the flowchart of the prototype generation algorithm unit. Prior to the cluster analysis, an initial step is to separate the input data into two subsets according to the temperature (>0° and <0°C). This is useful because the polarimetric signatures of liquid-phase and ice-phase hydrometeor types are very similar in some circumstances. However, in the classification step (Part II) we allow liquid-phase hydrometeors above the 0°C isotherm altitude and ice-phase hydrometeors below this altitude.

Fig. 3.
Fig. 3.

Flowchart of the prototype generation unit.

Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-13-00178.1

The clustering procedure is then performed on the data that have been sorted by temperature. In the first step, as shown in the flowchart of Fig. 3, a modified version of the K-means algorithm is applied with no initial conditions on the cluster means. The results are used as a prior for the expectation–maximization (EM) clustering (Dempster et al. 1977; McLachlan and Peel 2000; Hastie et al. 2009) based on Gaussian mixture models (GMMs). Each cluster is then validated using existing knowledge of polarimetric signatures, and each validated cluster is promoted as a “prototype.” All prototypes derived from multiple radar volumetric scans are finally used to derive the PDFs of polarimetric variables and temperature for each hydrometeor type.

Some issues need to be considered and solved in such techniques, such as the determination of the number of clusters, the criterion for promoting a cluster to a prototype, and the management of all prototypes derived from several radar volumetric scans. In the next subsections, we describe the clustering method and the solutions to address these issues.

a. K-means clustering

The K-means clustering is one of the most widely used clustering methods (e.g., Jakob et al. 2005; Marzban and Sandgathe 2006; Fereday et al. 2008; Caine et al. 2009; Chéruy and Aires 2009). In a similar vein to the work of Wilks (2011), the K-means algorithm includes the following steps:

  1. Compute the mean for each cluster.

  2. Calculate the Euclidean distances between a data point and the cluster means.

  3. If this point is already a member of the cluster whose mean is closest, repeat step 2 for the remaining data; otherwise, reassign this point to the cluster whose mean is closest and return to step 1.

In our implementation, a data vector of four polarimetric variables and temperature are normalized to a zero-mean and unity covariance matrix using Cholesky factorization. The normalization process is summarized as follows:

  1. Calculate the difference diff between each data point and the mean for each variable.

  2. Calculate the covariance diff of diff.

  3. Produce an upper-triangular matrix chol of diff using Cholesky factorization.

  4. Calculate the matrix division diff/chol.

The algorithm randomly selects K cluster means from the data vector as initials and repeats the clustering procedure N times to search for the result with the smallest sum of distances. In our experiment, we chose N = 5 because the attempts of more than five did not change the results in any obvious way.

As discussed in section 1, the PDFs of polarimetric variables and temperature overlap for some hydrometeor types (Bringi and Chandrasekar 2001; Doviak and Zrnić 2006). However, the K-means method produces hard boundaries between clusters, which are not compatible with overlapping PDFs. Therefore, a second clustering based on GMMs is adopted to convert the Euclidean distance into a probability. This can be considered as a refinement to the K-means algorithm. This method is described in the next subsection.

b. Expectation–maximization clustering based on GMMs

A cluster can be defined as data points belonging most likely to the same probability distribution in the clustering model. In the clustering based on GMMs (Fig. 3), the dataset is modeled as a Gaussian mixture consisting of a finite number of weighted Gaussian distributions whose parameters are iteratively optimized to best fit the dataset. If there are K distributions in a Gaussian mixture, then each Gaussian distribution is parameterized by mean and covariance , and the data are denoted by for n data points, (d = 5 in our case), . The kth distribution can be expressed as
e5
where the superscript T is the vector transpose. The Gaussian mixture is then given as
e6
where is the weight of the kth distribution. The log-likelihood function is defined as
e7

We adopt the EM algorithm to calculate the parameters of the K distributions (Hastie et al. 2009). The EM algorithm consists of two iterative steps: the E step and the M step. The E step assigns probabilities for data points with respect to every distribution of the mixture, while the M step recomputes the parameters of the distributions based on the updated probabilities. The algorithm includes the following steps:

  1. Initialize parameters according to the results of the K-means clustering.

  2. E step: compute the posterior probability Pr(i|k) at the pth iteration for all and ,
    e8
  3. M step: compute the parameters within each distribution of the mixture,
    e9
    e10
    e11
  4. Repeat the E step and M step until the convergence criterion (12) is satisfied,
    e12
    where is the maximum tolerance.

The K distributions under a GMM are a function of the Euclidean distance between the data point and the cluster means, if a scalar covariance matrix is assumed. Therefore, the EM clustering based on GMMs is a soft version of the K-means clustering, deriving probability rather than deterministic distances of points to cluster means.

c. Number of clusters and mixed-habit clusters

For the K-means and the EM algorithms, the number of clusters needs to be specified. One way is to measure a criterion, such as the Akaike information criterion (AIC) or the Bayesian information criterion (BIC), and then search for the minimum as the number of clusters increases. The number of clusters corresponding to this minimum is considered as the “optimal” number. However, when calculating the AIC and BIC using the CP-2 S-band radar data, increasing the number of clusters always resulted in a decreasing AIC and BIC. This may be due to modeling uncertainties because there may be no mathematical model available that the algorithm is able to optimize, and the assumption of Gaussianity is a rather strong one on the data. To set a cluster number, we simply choose a value that is believed to be larger than the number of hydrometeor types that can possibly be discriminated with polarimetric variables and temperature (10 clusters above the freezing level, 7 clusters below the freezing level). However, when using this conservative approach, we still obtained individual clusters that clearly included more than one hydrometeor type. We show an example of a well-behaved cluster and a “mixed habit” cluster in Fig. 4. Increasing the number of clusters did not change this fact. In our experiments, we systematically found that the mixed-habit clusters were always composed of a main peak with many data points and a number of well-defined secondary peaks but with a much smaller number of points (as in Fig. 5a). This fact often occurs on one or two dimensions out of the five possible dimensions. We speculate that this is the reason why increasing the number of clusters did not solve the issue of mixed-habit clusters. Based on this observation, a method has been developed to detect the local peaks and valleys in these histograms. The search strategy is to detect the highest/lowest point of the PDF, around which there are points lower/higher by a factor of 0.5 with respect to its surrounding points on both sides. Figure 5b shows the detected peaks and valleys for the histogram given in Fig. 5a. The clustering algorithm then quickly converges using the initial values obtained with this method. The resulting clusters are well defined around the identified peaks and valleys in the histogram (Fig. 5c).

Fig. 4.
Fig. 4.

Scatterplots of the (a) well-behaved cluster and (b) mixed-habit cluster.

Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-13-00178.1

Fig. 5.
Fig. 5.

The Zh histogram of the (a) mixed-habit cluster, (b) detected peaks and valleys of the histogram, and (c) clustering result.

Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-13-00178.1

d. Promoting a cluster to a prototype

An important component in the prototype generation is the procedure to promote a cluster to a prototype, as shown in Fig. 3. For each volumetric scan processed with the cluster technique, a procedure to validate each cluster is undertaken to determine the goodness of a cluster. Three requirements are defined as follows:

  1. Clear separation between clusters

  2. Minimum volume of clusters

  3. Maximal number of points around cluster means

It is to be noted that we assume that a well-behaved cluster always corresponds to a single hydrometeor type, which is the dominant type in the radar volume. However, more than one hydrometeor type may be present in a radar resolution volume—that is a general drawback of most hydrometeor classification techniques with the notable exception of the study by Dufournet and Russchenberg (2011). We will not be attempting to solve this problem in this paper.

The different clusters derived from the training radar dataset are then individually attributed to a prototype corresponding to a particular hydrometeor type. Once all volumetric scans of the training dataset used to characterize the PDFs of polarimetric variables are processed, we end up with a set of prototypes grouped by similarity. Each set of prototypes is then attributed to (or labeled as) a given hydrometeor type using existing knowledge of the typical ranges of values of the polarimetric variables and temperature in the literature (e.g., Table 2).

Table 2.

Ranges of polarimetric variables and temperature for different hydrometeor types. More detailed thresholds can be found in Straka et al. (2000).

Table 2.

Although the data are sorted by the 0°C isotherm altitude, we allow liquid-phase clusters at a small range above this altitude (e.g., −10°C) and ice-phase clusters at a small range below this altitude (e.g., 10°C). For example, if a cluster is found in this small range above the 0°C isotherm altitude and the other parameters are clearly indicative of rain, then the cluster will be labeled as rain. We also expect that a rain/hail mixture could occur below the 0°C isotherm altitude or at a small range above the 0°C isotherm altitude. Hail can be found either above or below the 0°C isotherm altitude. Therefore, the identification of hail depends solely on the statistics of the polarimetric variables.

The probability distribution of each cluster is used to generate a prototype, which is defined as with a weight . The parameters , and are obtained in Eqs. (9)(11) in the EM clustering. Figure 6 gives an illustration of this procedure. The left column shows a cluster and the associated histograms on each dimension. The corresponding prototype and the Gaussian curves (red) are plotted in the right column. In some circumstances, the Gaussian shape function does not fit the histograms very well (e.g., Figs. 6e,f). In this particular case, it may be due to insufficient samples to properly characterize the PDF. The polarimetric variables can still delineate different hydrometeor types, since the PDFs do not highly overlap in this region.

Fig. 6.
Fig. 6.

(left) A cluster and (right) the derived prototype for moderate rain, observed at 0624 UTC 16 Nov 2008. The projection of five-dimensional distribution is displayed in a panel of (a),(b) ZhZdr; (c),(d) ZhKdp; (e),(f) Zh; and (g),(h) ZhTenv. In the first column, the areas with data points less than 10% of the total points are screened out, and the color bars are the number of data points. In the second column, the black marks are the means of the prototypes, and the color bars represent the density of the prototype.

Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-13-00178.1

e. Managing prototypes

One problem with the cluster-based method is that it may not find many prototypes for the hydrometeor types characterized by a much lower frequency of occurrence. For instance, the frequency of occurrence of hail is much lower than that of rain. Therefore, the prototypes obtained from multiple volumetric scans need to be combined to produce the PDFs. The combination of the prototypes is achieved using a renormalization process based on the prototype weights and the number of points included in the datasets. Note the mean and covariance of a prototype remain unchanged during this process. For example, we obtain with weights () from a dataset involving N1 data points and with weights () from a dataset of N2 data points. Terms and can be combined into a single mixture model,
e13
The prototype weights are renormalized as
e14
e15
e16
e17

Therefore, the prototypes derived from multiple datasets can be managed in a single prototype set that satisfies . The underlying PDF derived by each set of prototypes can then be modeled by a Gaussian mixture and corresponds to a given hydrometeor type.

Another problem that arises is a tendency for the number of prototypes for some hydrometeor types—for example, ice aggregates and rain—to grow rapidly, because their frequency of occurrence in the troposphere is higher than that for other hydrometeor types. On the one hand, having a large number of prototypes can bring considerable information about the PDFs. On the other hand, if the algorithm simply follows the statistical model that it was based on, the number of prototypes may be too large to enable computation of the PDFs. To reduce the number of prototypes while retaining information from the original PDFs as much as possible, a Kullback–Leibler (KL) divergence approach (Runnalls 2007) was adopted. In this method, a pair selection criterion is used to calculate the upper bound of the KL distance, , which is defined as
e18
e19
e20
where i and j represent two different prototypes, and i and j are diagonal matrices, and a square unitary matrix ij is calculated by a simultaneous diagonalization procedure of covariance matrices (Rao 2009). The prototype pair with the smallest is merged using the following procedure:
e21
e22
e23

Figure 7 shows that the PDF with a reduced number of prototypes (Fig. 7b) is not significantly altered when compared with the original PDF (Fig. 7a), although it is derived from a smaller number of prototypes.

Fig. 7.
Fig. 7.

Original prototypes on (a) ZhZdr fields and (b) reduced prototypes. The colored marks are the means of the prototypes.

Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-13-00178.1

4. Probability density functions

In the cluster-based method, all well-behaved clusters are labeled and then promoted to prototypes. The prototypes under an identical class label may be used to approximate the PDF by constructing a Gaussian mixture. The rationale for using a Gaussian mixture is that a Gaussian mixture composed of a number of Gaussian distributions can approximate any shape with minimal bias. In this section, we study the statistical properties of polarimetric variables for each category of hydrometeors that has been identified by the clustering technique, including light rain, moderate rain, heavy rain, hail, rain/hail mixture, graupel/small hail, ice aggregates, melting ice, and ice crystals. Table 3 shows a comparison between the 95% ranges (defined by the 2.5th and 97.5th percentiles on the cumulative density functions) and the well-established classification boundaries in Table 2 synthesized from other studies by Straka et al. (2000). These comparisons and the location of the prototypes within the storms of our training dataset have been used to attribute each set of prototypes to a hydrometeor type. Although this is clearly a human intervention in the overall process, it does not alter much the “objective” aspect of the clustering technique claimed in the introduction, as this attribution is somewhat obvious once each set of prototypes has been derived.

Table 3.

The 95% ranges of polarimetric variables and temperature for hydrometeor types. The 95% range is defined by 2.5th and 97.5th percentiles on the cumulative density function. The estimates of larger than 1 are due to the signal processing technique used by the radar with alternating polarization.

Table 3.

The data used in this study were collected by the CP-2 S-band dual-polarization radar located near Brisbane. Our training radar dataset is made of 25 volumetric scans observed between 0618 and 0642 UTC 16 November 2008, between 1400 and 1454 UTC 19 November 2008, and between 0654 and 0748 UTC 24 March 2013. These include well-developed subtropical storms, which produced large and moderate rainfall, large hail, and damaging low-level gust winds and microbursts at ground level. It is important to include cases where hail was reported to make sure that hail and rain/hail mixture prototypes, which are critical for severe weather warning issues, can be produced by our clustering technique.

a. Rain

In the literature, the rain type is sometimes divided into six categories according to Zh (Bluestein 1992; Straka et al. 2000), including light rain (Zh < 30 dBZ), moderate rain (30 < Zh < 40 dBZ), heavy rain (40 < Zh < 45 dBZ), very heavy rain (45 < Zh < 50 dBZ), intense rain (50 < Zh < 57 dBZ), and extreme rain (Zh > 57 dBZ). In our method, we consider three rain classes, including light rain, moderate rain, and heavy rain. The rationale for including more than one rain type is to adopt the best estimator in different rain regimes (e.g., Bringi et al. 2009; Pepler et al. 2011; Pepler and May 2012).

In our experiments, 162 rain prototypes were identified by applying the clustering technique to the CP-2 polarimetric data collected at temperatures greater than −10°C. The prototype means and derived PDFs for the three rain categories are shown below (see Fig. 8). The PDFs are displayed using contours of probability in order to show the degree of overlap between these three rain categories. The overlapped areas over the high-dimensional feature space indicate that Zh, Zdr, and Kdp are powerful discriminators for these three rain categories, while and Tenv are very similar (i.e., no information content). The labeling of these prototypes as rain is obvious given the PDF of temperature at which these prototypes are found (see contours in Fig. 8 corresponding to the typical ranges given in Table 2), and the overall relationships between polarimetric variables. This is especially clear in the ZhZdr and ZhKdp relationships. The relative labeling of three rain categories is also easily achieved by using the relative reflectivity ranges of the three categories. Below we discuss in detail the PDFs obtained for each of these rain categories.

Fig. 8.
Fig. 8.

Comparison of the (a)–(d) prototype means and (e)–(h) PDFs between rain categories [light rain (green), moderate rain (blue), and heavy rain (red)].

Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-13-00178.1

1) Light rain

Figure 9 shows the projections of the high-dimensional PDFs constructed by 70 prototypes (black dots). The 95% range of Zh is between 6.1 and 42.8 dBZ, while the Zdr range is between −0.4 and 1.3 dB. For Kdp, the lower and upper values are between −0.4° and 0.4° km−1. The 95% ranges of and Tenv are between 0.70 and 0.99 and between 0.4° and 22.4°C, respectively. The Zdr and Kdp distributions are broader than the ranges suggested in Table 2 (reproduced in Fig. 9).

Fig. 9.
Fig. 9.

PDFs for light rain. The projection of the five-dimensional distribution is displayed: (a) ZhZdr, (b) ZhKdp, (c) Zh–ρhv, and (d) ZhTenv. The black dots represent the prototype means, and the thresholds plotted follow the ones in Straka et al. (2000). The key shows Rl: light rain, Rm: moderate rain, Rh: heavy rain, and R: rain.

Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-13-00178.1

2) Moderate rain

The projections of the PDFs derived from 55 prototypes are shown in Fig. 10. The 95% ranges of Zh, Zdr, Kdp, , and Tenv are from 23.7 to 48.4 dBZ, from −0.4 to 1.7 dB, from −0.3° to 1.1° km−1, from 0.90 to 0.99, and from 5.1° to 23.1°C, respectively. This PDF closely matches the thresholds given in Table 2, except for the Zdr lower bound.

Fig. 10.
Fig. 10.

As in Fig. 9, but for moderate rain.

Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-13-00178.1

3) Heavy rain

Figure 11 shows the projections of the PDFs composited from 37 prototypes. The 95% range of Zh is between 37.4 and 57.0 dBZ, while the lower and upper values of Zdr are 0 and 2.6 dB, respectively. The Kdp region ranges from −0.2° to 2.7° km−1. The ranges of and Tenv are from 0.95 to 0.99 and from 5.7° to 23.6°C, respectively. We note that the ZhZdr distribution for heavy rain is characterized by two peaks—one matches the expected boundaries from the literature for heavy rain, but the other exhibits lower Zdr values. This second peak in the joint distribution is consistent with the polarimetric signatures of graupel/small hail. However, this possibility has been clearly ruled out by inspecting the location within the storm (ice aggregates or not much ice above these regions, no graupel). These prototypes with high Zh and modest Zdr are actually associated with new convective developments embedded within an older and widespread stratiform area. We therefore speculate that the lower values of Zdr combined with relatively high Zh are due to high concentrations of small-size drops. This indicates that these specific situations could potentially be discriminated and labeled using the prototypes. However, more cases would need to be included in our training dataset in order to further explore this possibility.

Fig. 11.
Fig. 11.

As in Fig. 9, but for heavy rain.

Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-13-00178.1

b. Hail

The main polarimetric signatures of hail are large Zh but relatively small Zdr and Kdp because hail tends to tumble without a preferential orientation. Thanks to these marked polarimetric signatures, and the low frequency of occurrence (i.e., low number of prototypes found), it was obvious to attribute one set of prototypes derived from the clustering technique to hail, using the guidelines from Table 2.

Figure 12 shows the projections of the PDFs derived from the 12 obtained prototypes. The lower bound of Zh is 36.6 dBZ, which is notably lower than the threshold given in Table 2 (45 dBZ). The lower and upper bounds of Kdp and are similar to the thresholds in Table 2, but the Zdr range is narrower, which might be due to the utilization of limited amounts of volumetric scans in our training dataset.

Fig. 12.
Fig. 12.

As in Fig. 9, but for hail, where G/Hs: graupel/small hail, R/H: rain/hail mixture, and H: hail.

Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-13-00178.1

c. Rain/hail mixture

The main polarimetric signatures of the rain/hail mixture are large Zh, and the values of Zdr and Kdp are usually larger than those for pure hail because of the rain contribution to the polarimetric variables. Occasionally, enhanced Zdr and Kdp have been observed in melting hail due to a ring of water forming on the hail (May et al. 2001). Values of smaller than those of rain or hail are also expected.

Figure 13 shows the projections of the PDFs with 16 prototypes identified as rain/hail mixture. The lower limit of Zh is 42.4 dBZ, which is slightly lower than 45 dBZ as listed in Table 2. The ranges of Zdr and Kdp are between −0.1 and 3.4 dB and between −0.1° and 4.1° km−1, respectively. Although the lower bound for is 0.83, most values exceed 0.95. This is different from the thresholds in Table 2. The range of Tenv is between −8.9° and 21.8°C. Careful inspection of the location of a rain/hail mixture within the storm clearly shows that the incidences of a rain/hail mixture were always found below regions where hail was found above the 0°C isotherm altitude at the same time or on the earlier volumetric scans, as expected.

Fig. 13.
Fig. 13.

As in Fig. 12, but for the rain/hail mixture.

Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-13-00178.1

d. Graupel/small hail

The typical polarimetric signatures of graupel with a diameter ranging from 0.5 to 5 mm are very similar to those of small hail of a diameter between 5 and 20 mm. The separation between the two hydrometeor types is therefore challenging. In the results of our cluster analysis, we have not seen any clear indication that the polarimetric signatures of graupel and small hail could be discriminated. In other words, the cluster analysis shows that there is not sufficient information content in the polarimetric variables to discriminate these two hydrometeor types. This is consistent with the fact that most hydrometeor classification techniques do not attempt to separate them either. As a result, the PDFs derived from our cluster analysis are attributed to “graupel or small hail.”

We obtained 20 prototypes for graupel/small hail and the projections of the derived PDFs are shown in Fig. 14. The 95% range of Zh is between 24.2 and 59.1 dBZ, which is similar to the thresholds in Table 2. The upper bound of Zdr is 0.1 dB higher than the one in Table 2. Similarly, the upper bound of Kdp is 0.3° km−1 higher. The ranges of and Tenv are between 0.93 and 1.01 and between −48.9° and 0.7°C, respectively.

Fig. 14.
Fig. 14.

As in Fig. 12, but for graupel/small hail.

Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-13-00178.1

Figure 15 shows a comparison of the prototype means and the PDFs for the three categories that include graupel and/or hail. The overlap of the PDFs indicates that Zh and Zdr are good variables to discriminate these three categories, while Kdp can be used to discriminate the rain/hail mixture from the other two categories. There are no clear classification boundaries between these three categories using or Tenv.

Fig. 15.
Fig. 15.

Comparison of the (a)–(d) prototype means and (e)–(h) PDFs between graupel/small hail (green), rain/hail mixture (blue), and hail (red).

Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-13-00178.1

e. Small ice crystals and aggregates

Three more sets of prototypes were identified by the clustering technique. The labeling of these three sets of prototypes was again made using the relative polarimetric signatures of each set of prototypes, the location within the storms in our training dataset, and the guidelines provided by Table 2. In Fig. 16, we compare the prototype means and derived PDFs for these three categories. Terms Zh and Tenv are found to be the most powerful discriminators for these three hydrometeors. Melting ice occurs near and below the freezing level, as expected, which is different from the other two hydrometeor types. Small ice crystals are also the most frequent type of ice hydrometeors found at very low temperatures, as expected, while ice aggregates get more frequent at higher temperatures just above the freezing level.

Fig. 16.
Fig. 16.

Comparison of the (a)–(d) prototype means and (e)–(h) PDFs between ice crystals (green), ice aggregates (blue), and melting ice (red).

Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-13-00178.1

1) Ice aggregates

The typical polarimetric signatures of ice aggregates are intermediate to large values of Zh (10–40 dBZ in Fig. 17) and low values of Zdr and Kdp, reflecting the nearly spherical aspect of these particles and the fact that they are canting and tumbling.

Fig. 17.
Fig. 17.

As in Fig. 9, but for ice aggregates, where IC: small ice crystals, IA: ice aggregates, MI: melting ice, ICh: horizontally oriented crystals, and ICυ: vertically oriented crystals.

Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-13-00178.1

Figure 17 shows the projections of the PDF obtained from the 110 prototypes derived from the cluster analysis. The 95% range of Zh is between 13.8 and 38.8 dBZ, while the Tenv region is from −44.0° to 0.2°C. For , the lower and upper values are 0.91 and 0.99, respectively. The ranges of Zdr and Kdp are from −0.6 to 0.3 dB and from −0.2° to 0.5° km−1, respectively.

2) Melting ice

The Zh for melting ice is usually smaller than 45 dBZ. The Zdr and Kdp of melting ice are distinct from ice aggregates because the melting ice includes some amounts of liquid water. Together with the temperature, a low is a clear distinct signature of melting ice relative to ice aggregates.

Figure 18 shows the projections of the PDF using the obtained 23 prototypes of melting ice. The lower and upper bounds of Zh are 14.5 and 43.8 dBZ, respectively. The Zdr ranges from −0.7 to 0.5 dB. The lower bound for Zdr is smaller than that shown in Table 2. The range of Kdp is between −0.5° and 0.6° km−1. The range of is between 0.89 and 0.99, and that of Tenv is between −4.6° and 6.5°C. It is interesting to note that some prototypes for melting ice are found at environmental temperatures slightly above the 0°C isotherm altitude, as derived from the radiosounding nearest in time and space. This confirms the fact that the temperature variable should be used with caution when deriving the PDFs of individual hydrometeor habits and later when deriving the hydrometeor habit from these PDFs in classification techniques.

Fig. 18.
Fig. 18.

As in Fig. 17, but for melting ice.

Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-13-00178.1

3) Small ice crystals

The main polarimetric signature relative to ice aggregates are the smaller Zh, given the smaller particle sizes, and the larger variability of Zdr, because ice crystals can be prolates, oblates, or quasi spheres (Pruppacher and Klett 1996) and can have preferential orientation as a result of their shape.

The projections of the PDFs derived from the 106 prototypes obtained with the clustering technique for small ice crystals are shown in Fig. 19. Some prototypes are characterized by positive Zdr and others by small Zdr, as expected. However, there is no indication of vertically oriented or prolate ice crystals in our training dataset, which will require further studies including more cases. Those significantly smaller minimum values of than those reported in Table 2 are likely caused by the intermittent inclusion of radar noise measurements in the production of the prototypes. The classification of small ice particle habits using polarimetric variables is a relatively new area of research, as new observations of winter storms at higher radar frequencies are available (Elmore 2011; Schuur et al. 2012; Thompson et al. 2014).

Fig. 19.
Fig. 19.

As in Fig. 17, but for ice crystals.

Citation: Journal of Atmospheric and Oceanic Technology 32, 7; 10.1175/JTECH-D-13-00178.1

5. Summary

A new cluster-based method to derive the PDFs of polarimetric variables and the temperature of different hydrometeor types from polarimetric weather radar data is presented. This method consists of a two-step clustering procedure with K-means- and GMM-based EM clustering algorithms. The procedure iteratively operates with prior information obtained by searching peaks and valleys in data histograms to generate well-behaved clusters that can match data domain regions corresponding to certain hydrometeor types. Validated clusters are then promoted to prototypes. These prototypes are finally grouped in individual sets and each set is attributed to a hydrometeor type using guidelines from the literature. The potential of this technique is illustrated using a training dataset of 25 volumetric scans collected by the S-band CP-2 radar located in Brisbane, Australia. The prototypes obtained from the CP-2 polarimetric radar observations are used to derive the PDFs of polarimetric variables for each hydrometeor type. The analysis results show an agreement between the PDFs of polarimetric variables obtained using these prototypes and the ranges of values of the polarimetric variables suggested in Straka et al. (2000). However, some notable differences are also found. These differences could be attributed to the fact that the PDFs derived in this paper use all the polarimetric variables at once to solve a multidimensional problem using the clustering approach, whereas Straka et al. (2000) consider mainly pairs of independent variables, and these are used to delineate parallelograms regions.

As discussed previously, the cluster-based method has some advantages by definition over using ad hoc assumptions. It does not rely on preconceived ideas about the PDFs of the polarimetric variables corresponding to each hydrometeor type but derives objectively separable clusters from which PDFs of polarimetric variables can be constructed. The analysis of more radar data will be undertaken in order to refine the PDFs of polarimetric variables for the hydrometeor types identified in this first study and to attempt to discriminate more hydrometeor types. We will also investigate the potential of the clustering techniques to discriminate nonmeteorological echoes from meteorological echoes. To achieve this goal, we will consider more radar variables, including textures of reflectivity factor and differential phase shift.

The prototypes generated by the clustering method can now be used in hydrometeor classification methods using the CP-2 radar data. Two possible techniques (a prototype-level classifier and a distribution-level Bayesian classifier) are described in Part II of this study.

Acknowledgments

Michael Whimpey and Ken Glasson from the Australian Bureau of Meteorology provided assistance with the CP-2 radar data. The authors thank Tom Keenan and Vickal Kumar for their review of this work. We would also like to sincerely thank the four anonymous reviewers for their careful and helpful comments, which have greatly contributed to the quality of the paper.

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