Radar-based hydrometeor classification typically comes down
to determining the dominant type of hydrometeor populating a given radar
sampling volume. In this paper we address the subsequent problem of inferring
the secondary hydrometeor types present in a volume – the issue of
hydrometeor de-mixing. The present study relies on the semi-supervised
hydrometeor classification proposed by

Precipitation, and in particular snowfall, often occurs as a mixture of
several different hydrometeor types

Hydrometeor classification is a very popular topic in the weather radar
community, particularly since dual-polarization radar became a widely used
technology

In an effort to find a combination of these two conceptually opposed ideas, a
series of semi-supervised methods was proposed

Aside from assigning a label with the hydrometeor type to every volume,

The article also investigates the potential impact of residual incoherency in
weather radar measurements. This effect can be presumed to be likely in the
case of radar sampling volumes with mixed hydrometeors, despite the
conventional pulse averaging, which is supposed to filter out the contribution
of interferences. Namely, stronger random interferences between the
backscattered signals from the different hydrometeors populating a radar
sampling volume can be expected in the case of a more pronounced heterogeneity
among particles in a volume, potentially resulting in non-negligible residual
interference contribution. The study is done through the neighborhood-based
analysis, which is conducted by introducing the blind source separation (BSS)
techniques, principal component analysis (PCA) and independent component
analysis (ICA), into the weather radar data processing
(Fig.

Simplified schematic representation of

The article is organized as follows: in Sect.

The main objective of this paper could be summarized as drawing a parallel between the specific polarimetric framework of weather radar and the paradigm of decomposition/unmixing commonly used in SAR and hyperspectral remote sensing. The motivation for relying on the experience of SAR and hyperspectral communities comes from the demonstrated pertinence and utility of decomposition/unmixing in the data interpretation. This link we want to elaborate on, with the aim of de-mixing a weather radar sampling volume, is constructed around the common variable – entropy.

The radar variables we rely upon to discriminate between different
hydrometeor types are the reflectivity factor at horizontal polarization
(

Due to the skewness and the leptokurticity (fatter distribution tails) of
their distributions,

The multi-dimensional (four out of five dimensions) space:

Therefore, each radar sampling volume is characterized by a five-element
weather radar target vector,

The concept of entropy (

The min-entropy, being the minimum of Rényi's entropies

The estimation of probabilities

The bin-based approach considers one volume at a time, without taking the
neighboring spatial context into account. The proposed method is inspired by
the SAR coherent polarimetric decomposition of elementary processes (e.g.,
Pauli and Krogager,

Namely, the coherent decomposition of polarimetric SAR (PoLSAR) data relies
on the first-order statistics and represents the scattering matrix of a
target as a coherent sum of the scattering matrix of elementary interactions
(odd bounce and double bounce with two different orientations). By comparing
this to our polarimetric framework introduced in Sect.

An example inspired by

In terms of non-orthogonality, the hyperspectral linear unmixing problem
appears more analogous to our polarimetric framework (Fig.

When comparing Figs.

Now that we have identified our centroids as standard mechanisms (analogy
with PolSAR) and have found a mode for their addition via the distances in
the Euclidean space of the classification, we have to adapt the latter to the
particular nature of the former. That is to say, our centroids do not form a
regular structure (e.g., cube), where under the assumption of coherence and
linearity the distance of a measurement with respect to the standard
mechanism could be directly interpreted as the probability, i.e., as the
proportion of the given standard mechanism. They are rather non-uniformly
distributed in the five-dimensional space. In order to deal with this, we
adopted the varying slope exponential transformation of the distance between
the measurement and the

Namely,

An example of the exponential transformation: the scaled distances
to the probability (

The threshold value of probability (

It would be indeed even more precise to combine the mixing components at the level of the electromagnetic scattering, before the integration leading to the employed polarimetric parameters. However, not event this would help us overcome the unavoidable lack of methodological “transparency” in terms of physics, due to the absence of the orthogonal de-mixing basis. The proposed method is therefore rather defined in a more empirical fashion, in the data processing plane, with the physical trustworthiness being verified experimentally, mostly using the independent measurements.

Quadratically organized synthetic dataset used in the entropy
parametrization:

Applying the hydrometeor classification to the synthetic dataset results in
an expectedly proper recognition of pure hydrometeor boxes
(Fig.

The parameterized entropy, obtained by substituting

Classification and entropy applied to the synthetic dataset from
Fig.

The utility of the proposed parameterization of entropy, which is based on
the estimated proportions of different hydrometeors involved in a mixture, can
be adequately illustrated using the exemplary data introduced in
Fig.

The multi-dimensional (four out of five dimensions) space, with target
vectors representing centroids (larger points with different classes depicted
by different colors) and observations with the level of gray depicting

Before proceeding to the performance analysis, in order to quantify the
potential implicit biases of the introduced parameterization, we apply the
de-mixing method to the particular combinations of hydrometeors mixed in both
equal and non-equal proportions. Namely, as illustrated in
Fig.

Classification (1) and entropy (2) for different combinations of
synthetically produced mixtures:

Quantification of the results presented in Fig.

Quantitative evaluation of de-mixing errors (biases) obtained using synthetic (simulated) dataset.

The performance of the introduced bin-based method is analyzed hereby in three respective stages, very characteristic for the validation of techniques related to the hydrometeor classification: spatial plausibility, comparison between two radars and comparison between a radar and a ground level instrument.

In Fig.

The results of the bin-based de-mixing, illustrated in
Fig.

Figure

Bin-based de-mixing applied to an example MXPol dataset acquired
during the APRES3 campaign at the Dumont-d'Urville base, Antarctica, on
28 January 2016:

Bin-based de-mixing applied to an example MXPol dataset acquired
during the HyMeX campaign in the region of Ardèche, France, on
24 September 2012:

The following two stages of the performance verification are related to the
measurement campaign organized in the Swiss canton of Valais, from
November 2016 to April 2017. The campaign was based on the careful
collocation of different instruments, depicted in Fig.

Not having the possibility to retroactively manipulate the scanning strategy,
we decided to base the second step of the verification on analyzing the
influence of the proposed de-mixing method on the classification matching
between two radars covering a certain common volume. Namely, we defined the
vertical cross section sized 7 km in range and 2 km in height, being common
for the Plaine Morte radar 227

Configuration of instruments deployed during the Valais campaign.

One of the acquisitions, illustrated in Fig.

With the data being kept in polar coordinates, it was impossible to properly match
the volumes. Thus, a more direct, quantitative way of proving the utility of
the proposed de-mixing approach comes down to the comparison of the
proportions of detected classes in the entire cross section, before and after
the de-mixing (Fig.

In Fig.

The share of the dominant class (Fig.

We also benefited from this configuration checking for the potential
correlation between the entropy parameter/mixture indicator

Comparison example in terms of classification

The final stage of the verification is based on comparing the outcome of the
de-mixing method with the classification of individual particles from the
ground level instrument. The principle intuitively resembles the comparison
with classification based on the 2-Dimensional Video Disdrometer (2DVD)

The setup is again based on considering the vertical cross section of the
reconstructed RHI of the Plaine Morte radar, though this time in a slightly
more restricted area of 4 km in the range direction, around the MASC
(Fig.

In order to fully satisfy the hypothesis of stationarity (in terms of
proportions of dominant labeled hydrometeors), which allows us to properly
average the de-mixing scores, we selected different periods across different
events, summarized in Table

Comparison of classification applied to Plaine Morte data

Quantitative scores (

Though the quantitative evaluation of mismatching shows pretty good results
from the classification itself, similarly to the second stage of the
verification, we can still notice that in all six analyzed events, regardless
of their duration, the applied de-mixing method improves the distribution
matching (smaller

In Fig.

In order to demonstrate the capability of the method of dealing with the
hydrometeor classes other than the over-represented aggregated and rimed ice
particles, we illustrate the comparison of distributions across four classes
for event no. 5 in Fig.

Comparison of classification applied to Plaine Morte data with the
MASC classification, before de-mixing

The agreement with the MASC is further reinforced by considering the
additional parameter estimated from the MASC measurements – the continuous
riming degree index (DoR)

Radar vs. MASC, detection of rimed particles:

In Fig.

The neighborhood-based analysis is founded on simultaneously considering an
ensemble of pixels, rather than one pixel at a time as was the case with
the bin-based approach. This approach increases the potential issue of the
spatial incoherency in weather radar measurements

In the radar sampling volume populated by different hydrometeor types
(characterized by significantly different shapes and fall velocities), it is
logical to suspect that some residual interferences could “survive” the
conventional averaging over several pulses. Embracing the hypothesis of
originally incoherent measurements, one could consider this to be the
residual incoherency, though this could equally be an intrinsically
coherent backscattering as described by

This being said, we decided to proceed with the following analysis, inspired
by SAR incoherent polarimetric decompositions

It should be noted that the phase indicator, as external information, is not
included (phase indicator, the fifth element of our weather radar target
vector (Eq.

Principal component analysis is a statistical method which transforms
the data represented in a space formed by correlated variables to the space
formed by orthogonal, linearly uncorrelated variables

Going backwards, by applying the inverse PCA transform, we can estimate the
proportions of the originally measured samples contributing to each of the
pure uncorrelated components:

Independent component analysis allows for a more rigorous separation of
components with respect to PCA

Their independence is reached by relying on the paradigm used in PCA
(eigenvalue decomposition) but applied to tensorial structures, which are
higher-order generalizations of covariance matrices. Alternatively, it is
done by means of an iterative process aiming to increase the non-Gaussianity
of the sources. In the latter case, adopted in this analysis, the hypothesis
is that, due to the central limit theorem, the increase in the non-Gaussianity
of the sources will lead to the increase in their mutual independence

As was the case with PCA (Eq.

After elaborating in the previous section the de-mixing of aggregates, rimed
ice particles and crystals through the bin-based approach, now we approach
the same problem by trying to evaluate the potential lack of coherency. As
previously stated, the polarimetric parameters characterizing a radar volume
(

The example of (1) MXPol radar RHI and (2) Plaine Morte
reconstructed RHI (from the dataset used in Sect.

In Fig.

We start by taking all the pixels observed by the MXPol radar in one of the
acquisitions during the extended event introduced in
Sect.

PCA (blue) and ICA (red) applied to the

As suggested in Sect.

The distribution of the proportions of the most dominant component,
calculated by generalizing the analysis illustrated in Figs.

Unfortunately, by evaluating the entropy estimate of
the pure component in the space of the original centroids, we see that these
vectors

Aside from uncorrelated components, in Fig.

PCA (blue) and ICA (red) applied to the

The same analysis applied to the Plaine Morte data (24 acquisitions between
4:00 and 6:00 UTC on 28 February 2017) is illustrated in
Fig.

Aside from studying the potential effect of incoherency, this analysis is
also useful in highlighting the limit of the concept of discrete hydrometeor
classification. Namely, this concept prevents us from exploiting the
conventional tools in dealing with the residual incoherency that could allow
us to have an even more systematic and assumptions-free insight into the
mixed-radar sampling volumes, with respect to the one presented in this
article as the bin-based approach. A possible way forward would be to
investigate the possibility of modeling the weather radar target vector.
Following suggestions from the micro-physical modeling
community

In this paper, we address the issue of hydrometeor mixtures in polarimetric radar measurements by adapting the paradigm of decomposition/unmixing widely elaborated in other remote-sensing domains to the field of weather radar remote sensing.

In the first part of the paper we propose a bin-based de-mixing approach,
which is largely based on the hypothesis of coherent backscattering of
hydrometeors inside the radar sampling volume. The proposed approach is built
upon the semi-supervised hydrometeor classification method which reduces the
classification problem to the distances in the Euclidean space formed
essentially by the polarimetric parameters

The second part of the paper is dedicated to the study of a potential influence of the residual spatial incoherency in the backscattering of hydrometeors inside the radar sampling volume. The study is based on adapting the conventional statistical methods, such as PCA and ICA, used to deal with the spatial incoherency in the SAR remote sensing to the specific framework of the weather radar polarimetry. The performance analysis points out the limited influence of the residual incoherency in the regions of hydrometeor mixtures. The introduced evaluation of the spatial consistency in the case of heterogeneousness radar sampling volumes is important given that potentially present incoherency is not only due to the intraclass variability but also due to the interclass hydrometeor variability. The conclusion, implying that after all there is not a significant rise in incoherency in the case of hydrometeor mixtures on the one hand strengthens the proposed bin-based approach and on the other hands makes the tools such as PCA and ICA less useful in the context of weather radar decomposition/de-mixing than they are in the context of SAR remote sensing.

The overall message of this paper is to focus some attention of the weather radar community on the importance of the decomposition/de-mixing methods, which make it possible to look into the radar sampling volume. The present work remains exploratory, and many avenues still need to be explored, including the potential benefits of a continuous hydrometeor classification approach. Finally, the proposed bin-based approach, allowing already plausible and fairly validated estimation of hydrometeor type at the sub-bin level, can be used to improve the quantitative estimation of precipitation using radar.

Codes can be made available upon request to the authors. Datasets acquired by the MXPol X-band radar and the MASC can be made available upon request to the authors. For data acquired by the operation C-band radar network, contact the authors affiliated with MeteoSwiss.

NB and AB developed the concept of the paper, performed the analyses and interpreted the results. CP and JG particularly contributed to the aspect related to MASC acquisition and processing. JFV and JG notably contributed to the radar data acquisition and processing part. UG and MG notably contributed to the conceptual and the interpretation segments. NB, with contributions of all authors, prepared the manuscript.

The authors declare that they have no conflict of interest.

With great appreciation of their very useful comments and suggestions, the authors would like to thank the five anonymous reviewers. The authors would also like to thank their colleagues at LTE and the Radar, Satellite and Nowcasting teams for all their useful suggestions and their support in the data processing. Particularly, we would like to emphasize the help of Floortje Van den Heuvel and Peter Speirs due to their indispensable role in the organization of the campaign of measurements in the canton of Valais. Edited by: Gianfranco Vulpiani Reviewed by: five anonymous referees