# Extended Polarimetric Observations of Chaff using the WSR-88D Weather Radar
Network

James M. Kurdzo, Betty J. Bennett, John Y. N. Cho, and Michael F. Donovan All
authors are with the Massachusetts Institute of Technology Lincoln Laboratory,
Lexington, MA 02421 USA email<EMAIL_ADDRESS>
###### Abstract

Military chaff is a metallic, fibrous radar countermeasure that is released by
aircraft and rockets for diversion and masking of targets. It is often
released across the United States for training purposes, and, due to its
resonant cut lengths, is often observed on the S-band Weather Surveillance
Radar – 1988 Doppler (WSR-88D) network. Efforts to identify and characterize
chaff and other non-meteorological targets algorithmically require a
statistical understanding of the targets. Previous studies of chaff
characteristics have provided important information that has proven to be
useful for algorithmic development. However, recent changes to the WSR-88D
processing suite have allowed for a vastly extended range of differential
reflectivity, a prime topic of previous studies on chaff using weather radar.
Motivated by these changes, a new dataset of 2.8 million range gates of chaff
from 267 cases across the United States is analyzed. With a better
spatiotemporal representation of cases compared to previous studies, new
analyses of height dependence, as well as changes in statistics by volume
coverage pattern are examined, along with an investigation of the new “full”
range of differential reflectivity. A discussion of how these findings are
being used in WSR-88D algorithm development is presented, specifically with a
focus on machine learning and separation of different target types.

###### Index Terms:

Weather radar, polarimetry, chaff.

##  I Introduction

Chaff, a military radar countermeasure that consists of metallic fibers cut to
specific resonant frequencies, is used to mask aircraft, ships, and missiles
from enemy detection [1]. While typically used operationally in the
battlefield, chaff releases are common in United States airspace, likely for
testing and training purposes (e.g., [2]). Given observations of chaff lengths
ranging from 1–20 cm on the ground following releases [2], these metallic
dipoles resonate at frequencies easily detectable by the 10-cm wavelength
Weather Surveillance Radar – 1988 Doppler (WSR-88D) network across the United
States [3, 4]. An example of chaff fibers and clumps is shown in Fig. 1 (from
[5]).

Chaff is an air motion tracer, meaning that it often flows with the underlying
atmospheric velocity field [6, 7, 8, 9, 10, 11]. While this can theoretically
be useful to users of weather radars, chaff is generally considered clutter
for these systems [12, 13, 14]. In many cases, chaff can be nearly co-located
with convective storms, and can take on an appearance that makes it
exceptionally difficult to differentiate from weather. This is especially true
for radar users that are either inexperienced or do not have access to
polarimetric radar variables, which can greatly aid in differentiating between
clutter and weather. As an example, air traffic controllers fall into the
latter category, since their displays primarily offer horizontal reflectivity
factor ($Z_{H}$) and not polarimetric data [15]. An example of chaff mixed
with weather is shown in Fig. 2. The areas of chaff are nearly impossible to
identify by $Z_{H}$ alone, but become discernible in the polarimetric fields,
especially differential phase ($\phi_{DP}$).

As part of efforts to characterize chaff and sea clutter (another clutter
target for weather radar), several statistical properties of chaff have been
examined across hundreds of cases for use in machine learning (ML)
classification algorithms. These properties include $Z_{H}$, differential
reflectivity ($Z_{DR}$), co-polar correlation coefficient ($\rho_{HV}$), and
$\phi_{DP}$. Statistical chaff characteristics were first presented by [16],
but only one case in central Oklahoma was examined, leading to a long-held
belief that chaff primarily presented only as horizontally oriented fibers,
leading to mostly positive $Z_{DR}$. However, [5] showed more-detailed
statistics covering 2.2 million range gates of chaff data, resulting in a
deeper understanding of negative $Z_{DR}$ in chaff, how often it occurs, and a
theoretical basis for why it occurs.

Figure 1: Chaff of various cut lengths (from [5]). Figure 2: An example case
of chaff mixed with convection from the KBYX (Key West, FL) WSR-88D on 8
August 2016. Clockwise from the top left are horizontal reflectivity factor
($Z_{H}$), differential reflectivity ($Z_{DR}$), differential phase
($\phi_{DP}$), and co-polar correlation coefficient ($\rho_{HV}$)

The motivation for this study is an upgrade to the WSR-88D that expanded the
available $Z_{DR}$ range from -7.9 – 7.9 dB to -13 – 20 dB. Since a large
focus of [5] was on new observations of $Z_{DR}$ in chaff, it is important to
characterize $Z_{DR}$ with the new, expanded scale. While exploring these data
in a new, approximately 2.8-million range gate dataset collected in 2022,
several additional statistical characteristics in chaff that were not covered
in [5] were discovered. Additionally, through the development of an
upgraded/combined sea clutter and chaff detection algorithm, it was found that
the currently operational chaff detection algorithm (CDA) [17] with accuracy
of approximately 80% was able to be improved to nearly 95% accuracy with the
statistics gleaned from this new database when combined with similar data
collection of sea clutter cases [18]. This paper focuses not only on the newly
observed $Z_{DR}$ statistics, but also on new observations of other
parameters.

This paper is designed to build upon, and augment, [5] for use as a training
tool for weather radar users across the weather enterprise. In addition,
ongoing efforts to detect and differentiate chaff and sea clutter [17] can
heavily benefit from these new statistics. Use of the expanded $Z_{DR}$ data,
as well as discussions of $Z_{H}$ and vertical reflectivity factor ($Z_{V}$)
dependence on height, an approximation of circular depolarization ratio
(pseudo circular depolarization ratio; $PCDR$), and different $Z_{H}$ and
$Z_{DR}$ characteristics by WSR-88D volume coverage patterns (VCPs) discussed
in this paper will be beneficial to ML algorithms currently under development
to discriminate chaff and sea clutter from weather, ground clutter, and
biological scatterers. Although the paper is focused on data from the WSR-88D
network, it is anticipated that the methods and application to ML algorithms
will be useful for other radar systems and networks around the world.

The paper is organized as follows: the data, methods, and description of
polarimetric weather radar variables are first discussed. Next, statistical
observations of chaff across the entire dataset are presented. Finally, a
discussion of the findings and conclusions is provided.

##  II Dataset

The dataset used in this study was collected during an intensive observation
period from January – June 2022 and includes 267 cases (including all tilts
from each case) and approximately 2.8 million individual range gates of chaff.
A “tilt” is a constant-elevation-angle scan, otherwise known as a plan
position indicator (PPI). A “case” is defined as a single volumetric scan at a
single point of time from one radar site during a chaff “event.” Events are
defined as an entire chaff release episode, which can sometimes last over 12
hours. Cases of chaff sometimes exist from separate radar sites at a similar
time to another radar site(s), but, in most cases, only one time per event per
radar is included in the database. A small number of events (fewer than 10)
contain two cases that are at least 8 hours apart due to significant changes
in chaff structure and statistics over a long-duration release. This means
that, for the majority of events, the dataset does not contain multiple
volumes from the same radar in the same event, making the database
spatiotemporally diverse, especially compared to [5]. In [5], some cases were
from similar times within the same event and from the same radar.
Additionally, in the present dataset, there is a significant increase in the
diversity across radar sites spatially. Cases from 71 sites across the CONUS
are included, with the most from one radar being 32 from KNKX (San Diego, CA).

It should be noted that despite the much larger number of cases in this paper
compared to [5] (267 versus 75), only a modest increase in the number of range
gates used for analysis is realized (2.8 million versus 2.2 million). This is
due to the fact that a wider range of sites and “types” of cases were included
in the dataset. In [5], many of the cases were relatively large spatially
since they were straightforward to identify. The currently operational CDA was
developed with the statistics from [5] and operationalized on the WSR-88D in
2021 [17]. This algorithm was used only to identify cases across the entire
WSR-88D fleet, including smaller spatiotemporal cases with varying statistics.
It is important to note that while the existing CDA was used to identify
possible cases, all data in this study were collected manually via the
description below. No automation of case selection was used.

The spatial distribution of sites and the temporal distribution across six
months in the present dataset makes it an arguably better representation of
chaff characteristics across the CONUS than that in [5], while also including
the added $Z_{DR}$ range, discussion of $Z_{V}$, new $PCDR$ calculations, and
separation by height and VCP. All statistics are calculated using the Open
Radar Product Generator Simulator (ORPGSim), a MATLAB clone of the operational
WSR-88D ORPG. ORPGSim provides nearly identical results to the operational
ORPG for generating products, but offers the flexibility of simplified
processing and workflows for collection and analysis of large amounts of data.
As part of this software package, a region-of-interest tool was used to
manually truth range gates of chaff using a subject matter expert-defined
polygon drawn around clusters of chaff at each tilt within a volume. A
description of this collection technique is detailed in Section 2c of [5].
Every tilt of every case was manually truthed by a human, and the associated
masks were used to isolate chaff range gates. It should be noted that there is
no actual ground truth of chaff in any of our cases (i.e., no definitive proof
that what we are seeing is indeed chaff). The only known case with literal
ground truth was presented in [2], where chaff was found on the ground after a
very concentrated chaff release. Our approach is based on the fact that no
other known target type displays these signatures, and the signatures that
serve as our “truth” are relatively well-accepted in the broader literature.

In order to further refine the chosen range gates, all $\rho_{HV}$ values
$\geq$0.97 were removed from the final dataset. Additionally, all range gates
with a standard deviation of $\phi_{DP}$ less than 10∘ were removed. The
standard deviation of $\phi_{DP}$ in this context is an ORPG-defined product
that uses unwrapped $\phi_{DP}$ values. This is a different approach from [5],
and is meant to remove range gates of chaff that are mixed with, and dominated
by, weather. The values chosen for censoring are based on the fuzzy logic
membership functions in the Inanimate class of the Hydrometeor Classification
Algorithm (HCA; [19]) discussed in [20]. These values were originally
determined using a genetic algorithm to optimize the membership functions
using a series of chaff, sea clutter, and ground clutter cases. They were
found to be the optimal combination of parameters to identify these target
types and avoid categorizing weather and other clutter as chaff or sea
clutter. These more-stringent restrictions also account for a non-trivial
decrease in the number of range gates in the database used for analysis.

Similar to [5], an unfolded “full” $\phi_{DP}$ product was mimicked in
ORPGSim, allowing $\phi_{DP}$ values up to 360∘. Values 60∘ and below were not
used due to the system initial phase generally being set to 60∘, resulting in
a large spike where $\phi_{DP}$ values at or below 60∘ are cut off. This
invariably misses some folded $\phi_{DP}$ values, but the number of range
gates that fit into this category is exceptionally low. For height
calculations, a standard 4/3 Earth’s radius model is used to calculate the
beam height at each range gate based on range and elevation angle [21].

In addition to the aforementioned radar variables ($Z_{H}$, $Z_{DR}$,
$\rho_{HV}$, $\phi_{DP}$), also presented in [5], this study also discusses
$Z_{V}$, Doppler spectrum width ($W$), and $PCDR$. The formulation used for
$PCDR$ is an approximation of the circular depolarization ratio ($CDR$) for
simultaneous transmission/reception of horizontal/vertical polarization (SHV)
radars defined in [22]:

$PCDR=\frac{1+Z_{DR}^{-1}-2\rho_{HV}Z_{DR}^{-1/2}}{1+Z_{DR}^{-1}+2\rho_{HV}Z_{DR}^{-1/2}}$
(1)

It is important to note that in this study, the inputs of $Z_{DR}$ and
$\rho_{HV}$ to (1) are in derived product form, as with the rest of the
analyzed data. Within the WSR-88D processing scheme, derived products consist
of post-processed “products” that are outputs from the ORPG; derived products
are the input source to most downstream ORPG algorithms such as the HCA. The
primary derived products are modified versions of the base radar variable
estimates, including processing such as recombination from 0.5∘to 1.0∘and
other data quality assurance measures.

$CDR$ has been shown to be useful for discriminating between meteorological
and non-meteorological targets [23, 22, 24, 25, 26] and, most relevantly to
this study, between hydrometeors and chaff [9]. Given that it is not a
“native” variable in SHV radars, it is rarely used operationally [22] and is
an unused tool in algorithmic development on the WSR-88D. Although [27] showed
that $CDR$ can be expressed in terms of linearly polarized radar parameters,
[9] noted that this formulation is quite sensitive to propagation effects, and
instead proposed $PCDR$ (or what they called depolarization ratio), which is
more robust against propagation effects. With work underway to differentiate
chaff from other target types, a discussion of $PCDR$ and its relation to
other variables of interest is considered warranted in this study.

##  III Statistical Observations

This section focuses on the broad statistical properties of chaff within the
newly collected dataset from 2022. A deeper discussion on selected topics is
presented in the following section. For each figure shown, all 2.8 million
range gates across all 267 cases are included in the histograms. Direct
comparisons are made with the dataset in [5] when appropriate.

Figure 3: Histograms of approximately 2.8 million range gates of chaff from
267 cases. From top left, counterclockwise: horizontal reflectivity factor
($Z_{H}$), spectrum width ($W$), differential phase ($\phi_{DP}$), and co-
polar correlation coefficient ($\rho_{HV}$).

Histograms of $Z_{H}$, $W$, $\rho_{HV}$, and $\phi_{DP}$ are shown in Fig. 3.
$Z_{H}$ shows a roughly Gaussian shape centered just below 10 dBZ. The
observed distribution differs from the “plateau” seen in Fig. 3 of [5], but is
centered in a similar range. The higher concentration of values from -10 dBZ –
10 dBZ is thought to be due to the addition of more cases with varying spatial
extents. As opposed to the dataset in [5], which chose cases that were often
at their peak spatial extent (i.e., the most easily identifiable cases), the
present dataset includes a much-higher number of cases that have relatively
small spatial footprints, both vertically and horizontally. Anecdotally,
through extensive data collection and observation, cases with smaller chaff
cloud sizes tend to have lower reflectivity, which lends credence to the
observed histogram.

The histogram of $W$ was not included in [5], but is shown here due to its
importance in the work from [20], which utilized statistics from [5] to
develop a new hydrometeor class (called “Inanimate”) for the WSR-88D HCA. It
was found in [20] that exceptionally low values of $W$ were present in all of
the target types encompassed by the Inanimate class, which include chaff, sea
clutter, combustion debris, and radio frequency interference. However, in
recent attempts to separate chaff from sea clutter, there are distinct
differences in the peak and tail of $W$, making these findings relevant at the
current time. It makes intuitive sense that chaff would have a low $W$
distribution since most cases collected did not overlap with convective storm
cells where high $W$ values can be expected. Increased values of $W$ do seem
to be present in cases of chaff near or adjacent to convection (not shown),
but the tail drops off quickly after 3 – 4 m/s. It should be noted that most
cases in the database consist of chaff-only scatterers, so the true values in
the $W$ histogram may extend slightly higher in cases where chaff is mixed
with convective weather. However, in most cases, the dominant scatterer (be it
chaff or weather) is what is “detected” by the radar. Of course, if chaff is
detectable but mixed with weather, we would expect to see elevated $W$
estimates.

In $\rho_{HV}$ and $\phi_{DP}$, the distributions are largely similar to those
shown in [5]. The differences are that $\rho_{HV}$ values greater than or
equal to 0.97 were discarded in this dataset due to the proximity of weather
in some cases, and values of $\phi_{DP}$ less than or equal to 60∘ were
discarded due to the setting of system initial phase. Range gates where the
standard deviation of $\phi_{DP}$ was less than or equal to 10∘ were also
discarded, which slightly alters the $\phi_{DP}$ distribution. The standard
deviation of $\phi_{DP}$ was calculated in the identical fashion to the ORPG,
which creates a “hidden” product that is part of the operational HCA. This
product is what is referred to as a texture field in [19], which is calculated
using a comparison between a sliding smoothing window and the original data.
The exact filter values are detailed in [19].

For $\rho_{HV}$, a peak near 0.5 with a relatively sharp dropoff toward 0.1 is
seen. A much more muted drop in distribution of $\rho_{HV}$ is seen towards
the highest values at 0.96. It should be noted that, as in [5], there are
cases of chaff range gates that extend to the maximum $\rho_{HV}$ estimate of
1.03, but they were discarded for this dataset. Note that in the WSR-88D
processing of derived products, some $\rho_{HV}$ estimates can be as high as
1.03. These values were clipped and removed in the histograms. For
$\phi_{DP}$, as with the dataset in [5], a peak around 100 – 120∘ is seen,
with a gradual dropoff toward 200∘ and a long tail to 360∘. Unlike Fig. 3 in
[5], folded values that end up below 60∘ are not included.

Typically, $\phi_{DP}$ is not a tool directly used in weather radar analysis;
rather, its path derivative, specific differential phase ($K_{DP}$), is used.
However, in many dual-polarimetric weather radars, the WSR-88D included,
$K_{DP}$ is not calculated for $\rho_{HV}$ values under a given value (0.9 for
the WSR-88D) due to the unlikelihood that rain is present. Therefore, on many
radars, $K_{DP}$ is not calculated in instances of chaff. It turns out that
$\phi_{DP}$ is itself very useful for identifying chaff due to its highly
stochastic nature compared with the relatively smooth gradients seen in
weather (as seen in Fig. 2). This appearance is due to the fact that chaff, if
treated as a dipole, has zero differential backscatter phase. The combination
of receiver differential phase as the primary contributor and high measurement
error due to very low cross-correlation coefficients [16] makes for this
hallmark stochastic presentation in phiDP [5]. In cases when the standard
deviation of $\phi_{DP}$ is less than 10∘, we have seen, both qualitatively
and quantitatively, that chaff is exceptionally unlikely to be present, or,
weather is the dominant scatterer in the resolution volume.

Figure 4: Histograms of approximately 2.8 million range gates of chaff from
267 cases. Left: differential reflectivity ($Z_{DR}$); right: pseudo circular
depolarization ratio ($PCDR$) from [22]. Vertical dashed black lines indicate
the previous $Z_{DR}$ range of -7.9 – 7.9 dB.

With the latest software processing upgrades to the WSR-88D (as of late 2021),
every case in the present 2022 dataset is able to provide $Z_{DR}$ information
between -13 – 20 dB. Previously, values that fell beyond -7.9 – 7.9 dB were
clipped. In [5], those clipped values were eliminated. In Fig. 4, the full
range of $Z_{DR}$ is shown with the dashed vertical black lines indicating the
previous -7.9 – 7.9 dB $Z_{DR}$ range. Within the previous range, the
distribution is a bit different than that seen in [5]. A more plateaued shape
is seen in the 2022 dataset, with relatively flat probability densities
between 0 and 5 dB. In [5], there was a peak around 1.5 dB and a slow dropoff
toward the 7.9 dB limit, while here, the peak is at 4 dB. The relatively
plateaued shape is in stark contrast to the sharper peak at 1.5 dB in [5],
demonstrating a consistent distribution of $Z_{DR}$ values between roughly 0 –
5 dB. This observation of a shift toward higher $Z_{DR}$ values is thought to
be indicative of a more-diverse dataset in the present study. For example,
with a wider spatial range of data (i.e., more sites across the country),
different types of chaff may be used, leading to a possible shift in
distribution. However, it is not completely clear why this shift is seen in
the 2022 data.

In Fig. 4, the dropoff after 5 dB is sharp and rapid, decreasing by nearly a
factor of two before reaching 8 dB. However, most importantly, a non-trivial
percentage of the distribution (17%) is observed outside of the original
bounds. This is particularly true on the positive side ranging from 7.9 – 20
dB. A total of nearly 15% of all chaff range gates fall above 7.9 dB.
Additionally, a small tail also extends from -13 – -7.9 dB, showing additional
information gleaned for strongly vertically oriented fibers with the new
$Z_{DR}$ range.

It should be noted that $Z_{DR}$ calibration is a known issue with the WSR-88D
system [28], leading to biases that can sometimes exceed $\pm$0.5 dB [29].
While these biases can be monitored using techniques such as Bragg scatter
detection, they are not recorded as a definitive value in WSR-88D data (and
can change over even short periods of time) [29]. Since correcting for
$Z_{DR}$ biases is impractical for this study, it is assumed that the biases
are small enough to still get the general trend of $Z_{DR}$ values. It is also
important to consider that the distribution of $Z_{DR}$ values in this study
represent the data used to train the updated CDA in [18]. This generally
favors algorithm performance, since calibration is not accounted for in real-
time data streaming either (i.e., as data are fed to the algorithm for
processing and display).

The distribution of $PCDR$, calculated using the approximation from [22] shown
in (1), has not been explored before in a large number of chaff cases. Several
studies have shown various types of clutter to fall within the range of -10 –
-1 dB, while weather is generally much more negative [22], since spherical
particles reflect much more strongly in the opposite circular polarization
sense than in the same sense. In Fig. 4, the $PCDR$ histogram fits this
expectation, with a peak close to -4 dB, a sharp dropoff to -1 dB, and a long
tail out to approximately -18 dB. Further discussion on these findings and
their importance to algorithmic development is provided in the next section.

Figure 5: Distribution of chaff range gates between horizontal reflectivity
factor ($Z_{H}$) and differential reflectivity ($Z_{DR}$). Horizontal dashed
white lines indicate the previous $Z_{DR}$ range of -7.9 – 7.9 dB.

Given the advent of a new range for $Z_{DR}$, it was prudent to compare
$Z_{H}$ versus $Z_{DR}$ in a similar fashion to Fig. 5 from [5]. The results
are shown in Fig. 5, with $Z_{H}$ along the abscissa and $Z_{DR}$ along the
ordinate. Similar to the differences in $Z_{H}$ distribution between Fig. 3
and [5], the distribution in Fig. 5 is skewed toward lower $Z_{H}$ values
compared to [5]. Additionally, the plateaued nature of the $Z_{DR}$
distribution in Fig. 4 is evident in Fig. 5. When combined, there is a range
of distinct hot spots from 5 dBZ / 5 dB (for $Z_{H}$ and $Z_{DR}$
respectively) to 15 dBZ / 0 dB (also for $Z_{H}$ and $Z_{DR}$ respectively).
While this is not surprising given the histograms in Figs. 3 and 4, two
additional interesting features are apparent. First, at $Z_{DR}$ values beyond
7.9 dB, the $Z_{H}$ values tend to sit between 0 and 20 dB. This indicates
that the extension of $Z_{DR}$ to higher values correlates with a wide range
of $Z_{H}$ values.

Figure 6: Height-based chaff distribution of horizontal and vertical
reflectivity factors ($Z_{H}$ and $Z_{V}$; top row, from left to right), as
well as differential reflectivity and pseudo circular depolarization ratio
($Z_{DR}$ and $PCDR$; bottom row, from left to right). Heights are relative to
the radar’s elevation.

In [5], the only examples of height dependence were presented for $Z_{DR}$ and
were focused on individual case studies. Violin plots were shown for several
ranges of heights spanning eight different case studies. $Z_{DR}$ was also
explored temporally throughout the lifetime of a chaff cloud’s fallout for the
same eight cases. However, no holistic view of $Z_{DR}$ with height was
explored, and no statistics of $Z_{H}$ were discussed in a spatiotemporal
sense. In Fig. 6, the dependence of $Z_{H}$, $Z_{V}$, $Z_{DR}$, and $PCDR$
with height are shown. Both $Z_{H}$ and $Z_{V}$ show trends of increasing with
height, although $Z_{V}$ peaks at slightly lower values at a similar height as
$Z_{H}$ (roughly 750 m above radar level). A discussion on this effect and
what it implies for algorithm development is provided in the following
section.

Also in Fig. 6, the dependence of $Z_{DR}$ on height is shown. In general, the
$Z_{DR}$ distribution above 250 m stays relatively similar with height, mostly
matching the histogram in Fig. 4 and agreeing with what would be expected from
the $Z_{H}$ and $Z_{V}$ distributions in the top row of Fig. 6. However, a
negative shift in distribution is apparent at lower heights, which would agree
well with the case studies in [5] where $Z_{DR}$ values close to zero were
seen at the lowest levels and $Z_{DR}$ values between 3 – 6 dB were more
common at higher elevations. Finally, $PCDR$ displays a relatively consistent
distribution with height, with a strong focus around -3 to -4 dB. There
appears to be a slight decrease in $PCDR$ with height, near -0.5 dB per km
between 250 m and 2 km. Additional discussion on these observations follows in
the next section.

Figure 7: Histograms of chaff horizontal reflectivity factor ($Z_{H}$) broken
out by precipitation and clear air volume coverage patterns (VCPs).

Finally, differences were noted in the database when comparing statistics
between different VCPs. VCPs are defined by the scan pattern of the radar,
including (but not limited to) the number/exact value of tilts, pulse
repetition frequency, and scan rate. In general, precipitation VCPs have more
tilts but scan at higher rates, resulting in both faster volumetric and
single-elevation update rates. Despite their name, clear air VCPs are often
used in cases of light stratiform precipitation, especially snow. VCP 31, a
clear air VCP, is a unique scanning mode for the WSR-88D in that it is the
only VCP that uses a longer pulse (4.7 $\mu$s) than the other VCPs (1.57
$\mu$s).

In order to investigate these observations, each range gate within each case
was separated into the current VCP at the time of the case. Histograms of
$Z_{H}$ for two sets of VCP combinations are detailed in Fig. 7: precipitation
versus clear air VCPs. In Fig. 7, the clear air VCPs show a much lower peak
$Z_{H}$ value in their distribution relative to the precipitation VCPs. The
precipitation VCPs are centered nearly 10 dBZ higher than the clear air VCPs,
but have a long, low tail out to -30 dBZ (compared with a more-gradual falloff
at low $Z_{H}$ values in the clear air VCPs). Thoughts behind these
observations are offered in the following section.

##  IV Discussion

A more-detailed discussion of three selected statistics is provided in this
section, including the distribution of $PCDR$, observations of $Z_{H}$,
$Z_{V}$, $Z_{DR}$, and $PCDR$ by height, and the distribution of $Z_{H}$ and
$Z_{DR}$ broken out by VCP.

###  IV-A Depolarization Ratio Distribution

As chaff is considered an air motion tracer, and is most-often found in
laminar airflow (i.e., not near convection), chaff is oriented horizontally
most of the time (as evidenced in Fig. 4). It is known from [5] that there is
a period of time during chaff fallout, usually during the beginning, that
features vertically oriented fibers, contributing to most of the negative
$Z_{DR}$ values in the histograms in both [5] and the present study.
Additionally, as discussed in [16], the linear depolarization ratio ($LDR$) in
chaff is expected to be relatively high compared to precipitation.

[16] assumed that the angle between the horizontal plane and the chaff axis
was distributed uniformly between zero and a maximum angle called the flutter
angle. With this approach, a flutter angle of zero corresponds to all chaff
being horizontally oriented ($Z_{DR}$ = +$\infty$), while a flutter angle of
90∘ corresponds to a uniform distribution of chaff orientation ($Z_{DR}$ = 0
dB). Their Hertzian dipole model using $LDR$ (their Fig. 3) showed that $LDR$
values are asymptotic approaching just above -5 dB as the flutter angle
approaches 90∘. In fact, the model only dips to about -10 dB at a flutter
angle of 30∘, which corresponds to an approximate $Z_{DR}$ of 15 dB. This
suggests that, in the presence of chaff, it would be expected to see values of
$LDR$ between -10 and -5 dB, but more extremely horizontally oriented chaff
could dip to lower $LDR$ values (-13 dB for a $Z_{DR}$ value of 20 dB).
However, $LDR$ cannot be measured by a radar operating in SHV mode, like the
WSR-88D. Thus, we consider the use of $CDR$ instead, which can be approximated
using SHV-derived variables via the $PCDR$ formulation.

For a radar operating in circular polarization mode, the $CDR$ is the received
cross-polarized power divided by the co-polarized power. $CDR$ is expected to
be an excellent discriminator between hydrometeors and chaff because a perfect
sphere reflects all signal in cross-polarized form ($CDR$ of +$\infty$),
whereas a resonant dipole (an approximation of a chaff strand) has a $CDR$ of
1 (0 dB). In reality, hydrometeors are not perfect spheres and chaff strands
are not perfect dipoles, but the $CDR$ for chaff is still typically expected
to be (and has been observed to be) about 20 dB greater than the $CDR$ for
most hydrometeors [9].

Radars that operate with SHV-polarized signals cannot directly measure $CDR$.
Although [27] showed that $CDR$ can be expressed in terms of linearly
polarized radar parameters, [22] noted that this formulation is quite
sensitive to propagation effects, and instead proposed the $PCDR$
approximation, or what they called depolarization ratio ($DR$), that is more
robust against propagation effects. $PCDR$, which is expressed via (1), is
computed from the measured SHV parameters that can be written as

$Z_{DR}=\frac{\left\langle\left|s_{\mathrm{hh}}\right|^{2}\right\rangle}{\left\langle\left|s_{\mathrm{vv}}\right|^{2}\right\rangle}$
(2)

and

$\rho_{HV}=\frac{\left\langle
s_{\mathrm{hh}}^{*}s_{\mathrm{vv}}\right\rangle}{\sqrt{\left\langle\left|s_{\mathrm{hh}}\right|^{2}\right\rangle\left\langle\left|s_{\mathrm{vv}}\right|^{2}\right\rangle}},$
(3)

where $s_{hh}$ and $s_{vv}$ are scattering coefficients for the horizontal and
vertical polarizations, respectively, and the asterisk denotes complex
conjugation.

Would $PCDR$ be as good a discriminator between hydrometeors and chaff as
$CDR$? For hydrometeors, [22] showed $PCDR$ to be generally less than -15 dB.
For chaff, one would expect $PCDR$ to be close to the theoretical 0 dB for
$CDR$, but it is not immediately obvious from (1) that this would be the case.
$Z_{DR}$ and $\rho_{HV}$ depend on the flutter angle of chaff [16]; although
the Hertzian dipole model is only strictly valid for chaff lengths much
shorter than the radar wavelength, [16] showed that its flutter angle
dependency is very similar to a thin wire model. The simple dipole model is
convenient, because a closed-form solution is available under reasonable
assumptions. A derivation of this model is provided in the Appendix.

This is visually illustrated in Fig. 8, where the resulting dependency of
$PCDR$ on the flutter angle is shown. It is clear that although $PCDR$ has a
flutter angle dependency, the range is quite limited (-3.7 to 0 dB). Since
hydrometeor $PCDR$ is expected to be generally less than -15 dB, this derived
SHV parameter seems promising for separating chaff from hydrometeor signals.
The $PCDR$ model result is substantially different than the $LDR$ model in
[16] in its limited range, but the histogram in Fig. 4 shows values much lower
(as low as -18 dB) than the minimum values shown in Fig. 8 (approximately -3.7
dB). We must consider the fact that chaff is not a perfect dipole, is
distributed in space, likely consists of inconsistent cut lengths, and can
clump [5], meaning that we can expect lower values in real-life scenarios.
Additionally, the model presented in (V) assumes a maximum flutter angle of
90∘, meaning that the orientation can only reach an equal distribution between
horizontal and vertical (i.e., $Z_{DR}$ of 0 dB). If the chaff orientation
distribution leans toward the vertical (i.e., negative $Z_{DR}$), the $PCDR$
values will extend lower than what is modeled in Fig. 8, likely partially
contributing to the extended tail seen in Fig. 4.

In considering the utility of the distribution shown in Fig. 4, the measured
distribution is still considerably narrower than that of hydrometeors, and the
majority of the distribution is well above the expected range of hydrometeor
$PCDR$ values [24]. This analysis suggests that an estimate of $PCDR$ can be
useful for discrimination of chaff using advanced methods such as ML. While an
ML algorithm could conceivably discern the different weights of importance of
$Z_{DR}$ and $\rho_{HV}$, especially in combination, between chaff and
weather, feeding more-primitive ML algorithms a direct relation such as that
provided in $PCDR$ may lead to a better/faster convergence on an acceptable
classifier. Indeed, this has been seen in the recent ongoing development of a
chaff detector, first presented in [17]. With the use of support vector
machines, overall performance has been determined to increase in their
approach by using a $PCDR$ estimate of WSR-88D data [18].

Figure 8: $PCDR$s for a dipole model based on flutter angle. Compared with the
$LDR$ results in [16], it is clear that the expected values of $PCDR$ are
higher than $LDR$ for a perfect dipole, even at extreme flutter angles. In
fact, the entire range of $PCDR$ is only about 3.7 dB for the a dipole
covering all flutter angles.

###  IV-B Height Dependence of $Z_{H}$, $Z_{V}$, $Z_{DR}$, and $PCDR$

Given the importance of driving a ML-based algorithm with relevant data that
can lead to convergence on a better model more quickly, it is relevant to
explore the spatial statistics of chaff. This is particularly relevant because
spatial variables (range, height, etc.) are readily available in real-time
processing, and any potential signal that can be derived using a large dataset
could be simply included in a ML algorithm (both in training and operations).
In order to identify these potential signals in the present dataset, four
variables are examined by height: $Z_{H}$, $Z_{V}$, $Z_{DR}$, and $PCDR$.
Relationships with range were also explored, but are not shown in this paper.

There is little reason to expect a direct dependence of chaff characteristics
as a function of range alone (other than a decrease in sensitivity), but there
are three other spatially oriented aspects that could theoretically have
impacts on what the radar estimates. These three aspects are height,
resolution volume size (which is correlated with range), and elevation angle
of the antenna. Height dependence could arguably be a physically viable
expectation due primarily to the results in [5]. For example, different stages
of chaff dispersal, as well as variations in the effect of the electric field
(e.g., by height and/or by chaff cut length) could have impacts on statistical
distributions. According to [5], chaff is often more likely to be vertically
oriented at lower elevations due to the increasing strength of the fair
weather electric field.

Resolution volume dependence could be viable because as the volume increases
in size, it is less likely that chaff will meet the volume-filled scattering
assumption in the calculation of reflectivity factor [21]. Finally, the
elevation angle of the antenna could have an impact due to the fact that the
“vertical” polarization of the radar becomes less truly vertical as the
elevation angle increases. This is because as the antenna points at higher
elevations, the angle of the vertical polarization becomes less orthogonal
relative to the ground (becoming parallel to the ground when pointing to 90∘).

In terms of viability, it seems unlikely that the impact of elevation angle
would be significant with the WSR-88D due to the relatively low antenna
pointing angles (up to 19.5∘ maximum, but focused mostly at lower angles,
especially in clear air VCPs). It also seems unlikely that the resolution
volume size is a meaningful factor, since we see the opposite effect in Fig.
6. That is, reflectivity increases with height rather than decreasing like we
would expect if the resolution volumes were not being adequately filled to
meet the reflectivity factor calculation assumption.

Therefore, we are left with exploring the impacts of height dependence based
on different stages of fallout. The subplots in Fig. 6 allow for the
exploration of these variations, including increasing $Z_{H}$ and $Z_{V}$ with
height, increasing $Z_{DR}$ with height, and a slightly decreasing $PCDR$ with
height. When chaff is initially released, it appears in a relatively small
footprint on the WSR-88D before dispersing with the prevailing atmospheric
wind. This dispersion results in size sorting vertically [5], with clumps and
vertically oriented fibers falling out prior to horizontally oriented fibers.
Before widespread dispersion spatially, the concentration of chaff is at its
highest point immediately post-release. Therefore, it makes intuitive sense
that $Z_{H}$ and $Z_{V}$ would be at their highest values at higher heights
(near release points), and decrease at lower elevations where the originally
concentrated chaff has dispersed significantly due to the effects of
differential advection (in the horizontal) and differential sedimentation (in
the vertical). Since sedimentation only operates in one direction (downward),
it makes sense that the further chaff gets from the height of release (further
downward), the weaker the reflectivity factor will be on average.

Additionally, the results presented in [5] suggest that lower $Z_{DR}$ values
are likely at lower heights (due to the impact of the fair weather electric
field), which agrees well with the $Z_{DR}$ distribution seen in Fig. 6. This
is because longer chaff strands are more-heavily impacted by the increasing
electric field strength closer to the Earth’s surface, causing them to tilt
more vertically and hence fall out faster. An additional consideration not
mentioned in [5] that has become evident in the current study is that there
may be holistic effects on $Z_{DR}$ as chaff descends into the planetary
boundary layer (PBL). Turbulent mixing within the PBL may change the chaff
orientation distribution from one that is (for example) mostly horizontal to a
more isotropic distribution, which would shift the $Z_{DR}$ distribution from
positive values toward values closer to 0 dB near the Earth’s surface.

One concern regarding the suggested PBL impacts is that there may be
unintended effects within the PBL in the chaff database. For example, if not
carefully considered, chaff in the presence of other target types typically
found within the PBL (clutter, insects, and other biota) could affect the
statistics. The truthing mechanism used in this study mitigates this concern
as much as possible. Given that each “cell” of chaff was manually traced using
a region-of-interest polygon tool, it was determined by a subject matter
expert that the gates being included were dominated by chaff. This is an
important distinction, since clearly chaff can be mixed with other scatterers.
In practice, if it was not exceedingly clear to the subject matter expert that
the dominant scatterer was chaff, the case was not included in the database.
While it is still possible that some biota and other targets are included, it
is expected that the influence is exceptionally small.

Finally, the slight decrease in $PCDR$ is difficult to explain physically. The
model depicted in Fig. 8 suggests that as the flutter angle decreases (a more
horizontally oriented distribution), $PCDR$ generally increases. This would
suggest that we should see increasing $PCDR$ with height since $Z_{DR}$ is
increasing with height. It should be considered that $\rho_{HV}$ is not being
taken into account when simply comparing against flutter angle (a $\rho_{HV}$
dipole model is not shown in this paper). Additionally, as mentioned above,
inconsistencies with chaff distribution, cut lengths, and clumps result in
$PCDR$ values well outside the range predicted by the simplistic Hertzian
dipole model. With these issues taken into account, it is not clear whether
the shift in $PCDR$ with height is statistically meaningful or if there is
even an underlying physical mechanism at play. A more-realistic model for
$PCDR$ in chaff would be required to determine the answers to these questions,
an endeavor that is beyond the scope of this paper. What is important in these
findings is that the distribution of $PCDR$ with height is remarkably narrow
and consistent, making this parameter an excellent candidate for chaff
identification.

One concern with the generalization of these statistics is that chaff is not
released in a systematic way. That is, releases do not seem to correlate
spatially, are not released in the same densities or at the same heights, and
of course, occur in different atmospheric conditions. As shown in [5], there
is certainly a case dependence, so it must be stated that large-scale
generalization has risks. However, after years of watching chaff releases on
the WSR-88D network, the authors have observed patterns in these releases that
are generalizable in, at least, qualitative terms. This is indicated by the
similarities in fallout patterns in the 8 cases shown in [5], which was the
impetus for the present study. Although qualitative generalization is helpful,
it is assumed that with millions of gates of chaff and hundreds of cases,
quantitative generalization is also both possible and helpful. In the case of
the data presented in Fig. 6, the discussion in this section presents several
theories for why these statistics are physically plausible.

These are generally unsurprising results given previous findings. However, the
implications cannot be understated. In a ML algorithm that does not take
height (or any spatial considerations) into account, comparing two areas of
statistical data at two very different heights could result in the same
inference, resulting in an incorrect classification. The findings in this
study helped the authors realize the importance of including spatial
dimension(s) in ML approaches to classification, as is described in [18]
compared to the original approach discussed in [17]. It should also be noted
that generalizations from a large dataset are necessary for effective ML
algorithm training. While these results generally match those from the case
studies in [5], case studies are not sufficient to train an expansive
algorithm. Discovery of these trends on a larger scale is important for such
classification problems.

###  IV-C VCP Dependence of $Z_{H}$

In searching for additional methods for improving the ML techniques used in
[17], the authors noticed that in several cases, the statistics in chaff
clouds seemed to change qualitatively when the VCP of the radar changed. This
led to the desire to a) quantify these changes, and b) determine the potential
applications for algorithm development. As with the importance of spatial
statistics mentioned in the previous subsection, changes in statistics by VCP
can cause incorrect classifications if not included in training an algorithm.
While qualitative statistical changes were observed for several variables, the
most quantifiable change with regard to large-scale statistics was seen in the
$Z_{H}$ histograms, which are presented in Fig. 7. The histograms for $Z_{H}$
are broken into two sets of VCPs: all precipitation VCPs and all clear air
VCPs.

Two possible explanations were brought to light during this analysis. First,
one of the clear-air VCPs, VCP 31, utilizes a longer pulse, leading to higher
sensitivity. However, this higher sensitivity only lowers the minimum
detectable $Z_{H}$, meaning the noise floor is effectively decreased. If VCP
31 was sufficiently represented in the database, and there were a large number
of gates that were below the noise floor of the other VCPs, it would be
plausible to think that VCP 31’s sensitivity could shift the distributions in
this way. A cursory analysis (not shown) of the spatial distribution of chaff
clouds in the database combined with an assumed chaff $Z_{H}$ distribution and
the performance metrics of each VCP showed that the shift in distribution was
negligible between the scan modes. In fact, the peak did not shift by more
than 1 dBZ, leading to the need for another explanation.

It turns out that the database is heavily weighted by clear-air observations,
since chaff is mostly observed away from weather. One of the hallmarks of
clear-air VCPs is that they only scan the lower tilts, leaving the higher
tilts out in favor of slower scan speeds while maintaining a respectable
volumetric update rate. When so many cases in the database only contain the
lower tilts, one must consider the height-based distributions in Fig. 6 that
show decreasing $Z_{H}$ at lower elevations above radar level. The difference
in precipitation versus clear-air VCPs in Fig. 7 are likely a manifestation of
the difference in height weighted by the percentage of cases collected in
clear-air VCPs.

It should be noted that there can also be other factors biasing the dataset in
this manner. We do not consider initial chaff release height, how long it
stays in fallout/dispersion, the distribution of heights/ranges versus the
available tilts, etc. There are far too many levers to pull that would be
potentially better explained by studies directed more at the physical
properties of chaff, such as T-matrix calculations. What is important here is
that ML algorithms, in their development, training, and testing/validation,
must take into account these differences in one way or another. In some cases,
the data used in training inherently include these biases; however, in the
case of chaff and weather radar, this is another example for the need for
spatial data modalities in model training and validation.

##  V Conclusions

In this study, an analysis of statistics in a new database of chaff cases was
completed using the WSR-88D weather radar network. A total of 267 cases were
used, spanning 71 sites across the contiguous United States (CONUS). This
dataset is unique in that it includes an expanded $Z_{DR}$ range from -7.9 –
7.9 dB to -13 – 20 dB. Additionally, several new parameters were investigated
that have not yet appeared in the literature, including a proxy for circular
depolarization ratio ($PCDR$), an analysis of $Z_{H}$, $Z_{V}$, $Z_{DR}$, and
$PCDR$ by height, and an analysis of $Z_{H}$ and $Z_{DR}$ statistics by VCP.
The anticipated physical mechanisms for these new observations were discussed,
and their relation to algorithm development, especially using ML techniques,
was focused upon.

The primary new findings are that $Z_{DR}$ values extend all the way to the
edges of the new $Z_{DR}$ range, but there is a wider spread in the histogram
at strongly positive values of $Z_{DR}$, especially between 8 and 13 dB.
$PCDR$ values peak at -4 dB, which is close to the value theorized in an
adapted $PCDR$-based model derived from the $LDR$-based Hertzian dipole model
in [16]. However, they trail off toward -20 dB, which indicates expected
differences between chaff and true dipoles. The separation relative to
hydrometeors is still sufficient to be an excellent separator between chaff
and weather, but it is not expected that it will help as much in
differentiating chaff from other statistically similar targets such as sea
clutter. While the results of $Z_{H}$ and $Z_{DR}$ distributions with height
are not necessarily surprising, they demonstrate the need to include spatial
variance in statistical distributions in radar algorithms, especially those
that use statistics in clusters with ML techniques. A similar conclusion can
be drawn using the VCP findings, which argue that different radar scanning
modes should also be included in ML algorithms.

It should be clearly stated that some of the discussion is speculative,
without full evidence for the hypotheses stated. While this may not be fully
scientifically satisfying, the hope is that the discussion will spur
additional research in these areas, specifically with the thinking that chaff
could actually help in making new scientific discoveries. A few examples would
be investigating PBL structure and electrification using controlled chaff
releases, as well as the potential to release chaff nearby supercell
thunderstorms in order to examine air motion in and around severe
thunderstorms [9].

Moving forward, in the future, the findings in this study should be
generalized to other radar systems, processing chains, and operating
frequencies. Application of the results in this study to non-WSR-88D radars is
recognized as a difficulty, since different radars/frequencies/calibrations
could have major effects on the distributions. Generalization would allow for
the ability to detect (and possibly filter) chaff across other radar networks.
There are multiple ways to accomplish this, including manually collecting
chaff cases on other networks/systems, or possibly by conducting T-matrix
scattering simulations in order to generalize the polarimetric characteristics
of chaff at multiple frequencies. It would also be useful to examine signal-
level (I/Q) data within chaff at multiple bands in order to determine any
specific spectral characteristics and/or attempt different processing
schemes/algorithms.

Finally, it would be prudent to investigate cases of chaff mixed with weather
as a specific focus in future work. Although this was not seen often,
distributions (and detection) in these cases would be useful to the community.
Building a database of these cases would be challenging and would likely need
to take place over multiple years (and with more time steps per case) due to
their scarcity. However, when chaff is located near weather, its contamination
is certainly a problem for weather radar users, making this an interesting
area for further exploration.

Specifically, spectral polarimetry is one potential method to separate chaff
from weather in a mixed resolution volume. This approach would require I/Q
data to explore, which is something that was not readily available for this
study. Future I/Q collection in chaff could open the door to exploring this
issue, although it is still not expected that clean separation can be achieved
since chaff and weather occupy the same Doppler spectral bins (i.e., chaff is
a passive tracer).

[Derivation of Hertzian Dipole Model for PCDR]

Assuming negligible multiple scattering and consistent chaff cut length, the
Hertzian dipole model yields [16]:

$\displaystyle\left\langle\left|s_{\mathrm{hh}}\right|^{2}\right\rangle=\left\langle\left|f_{b}\right|^{2}\right\rangle-\left\langle\left(f_{b}^{2}-f_{a}f_{b}\right)A_{2}\right\rangle+$
(4a) $\displaystyle\left\langle\left|f_{b}-f_{a}\right|^{2}A_{4}\right\rangle$

$\displaystyle\left\langle\left|s_{\mathrm{vv}}\right|^{2}\right\rangle=\left\langle\left|f_{b}\right|^{2}\right\rangle-2\left\langle\left(f_{b}^{2}-f_{a}f_{b}\right)A_{1}\right\rangle$
(4b)
$\displaystyle+\left\langle\left|f_{b}-f_{a}\right|^{2}A_{3}\right\rangle$

$\displaystyle\left\langle
s_{\mathrm{hh}}^{*}s_{\mathrm{vv}}\right\rangle=\left\langle\left|f_{b}\right|^{2}\right\rangle+\left\langle\left|f_{b}-f_{a}\right|^{2}A_{5}\right\rangle$
(4c)
$\displaystyle-\left\langle\left(f_{b}-f_{a}f_{b}\right)A_{1}\right\rangle-\left\langle\left(f_{b}^{2}-f_{a}f_{b}\right)A_{2}\right\rangle,$

where $f_{a}$ and $f_{b}$ are the scattering amplitudes for the electric field
along and perpendicular to the dipole axis, respectively. The $A_{i}$ values
are the scattering amplitudes assumed to be fixed with a transverse amplitude
of $f_{b}=0$. These amplitudes are closed-form solutions for the angular
moments as described in Eqs. 2–6 in [16]. Furthermore, if the scattering
amplitudes are assumed constant and the transverse component is zero ($f_{b}$
= 0), then (V) reduces to

$\displaystyle\left\langle\left|s_{\mathrm{hh}}\right|^{2}\right\rangle=\left|f_{a}\right|^{2}\left\langle
A_{4}\right\rangle$ (5a)

$\displaystyle\left\langle\left|s_{\mathrm{vv}}\right|^{2}\right\rangle=\left|f_{a}\right|^{2}\left\langle
A_{3}\right\rangle$ (5b)

$\displaystyle\left\langle
s_{\mathrm{hh}}^{*}s_{\mathrm{vv}}\right\rangle=\left|f_{a}\right|^{2}\left\langle
A_{5}\right\rangle.$ (5c)

Substituting (V) into (2) and (3) yields

$Z_{DR}=\frac{\left\langle A_{4}\right\rangle}{\left\langle
A_{3}\right\rangle}$ (6)

and

$\rho_{HV}=\frac{\left\langle A_{5}\right\rangle}{\sqrt{\left\langle
A_{3}\right\rangle\left\langle A_{4}\right\rangle}}.$ (7)

Substituting (6) and (7) into (1) gives

$PCDR=\frac{A_{3}+A_{4}-2A_{5}}{A_{3}+A_{4}+2A_{5}}.$ (8)

Assuming that the chaff is randomly oriented in the horizontal plane, the
radar antenna elevation angle is 0∘, and the angle between the chaff axis and
horizontal plane is uniformly distributed between 0∘ and a maximum angle
dubbed the flutter angle, [16] derive closed form solutions for angular
moments $A_{i}$ as a function of the flutter angle’s complement $\theta_{1}$
(i.e., the flutter angle is $\pi$/2 - $\theta_{1}$, and $\theta_{1}$ is
measured with respect to the true vertical):

$\displaystyle\left\langle A_{3}\right\rangle=\frac{1}{5}\cos^{4}\theta_{1}$
(9a)

$\displaystyle\left\langle
A_{4}\right\rangle=\frac{3}{40}\left(\sin^{4}\theta_{1}-\frac{4}{3}\cos^{2}\theta_{1}+4\right)$
(9b)

$\displaystyle\left\langle
A_{5}\right\rangle=\frac{1}{2}\left(\frac{1}{3}\cos^{2}\theta_{1}-\frac{1}{5}\cos^{4}\theta_{1}\right).$
(9c)

## Acknowledgment

The authors thank three anonymous reviewers for their helpful comments in
revising the original manuscript. This material is based upon work supported
by the Observations Program within the NOAA/OAR Weather Program Office under
Award No. NA21OAR4590385. Any opinions, findings, conclusions, or
recommendations expressed in this material are those of the authors and do not
necessarily reflect the views of the National Oceanic and Atmospheric
Administration. The authors would like to thank Autumn Millard for her
assistance with this work during her internship in the Summer of 2022.
Discussions with Earle Williams, David Smalley, Mark Veillette, David
Patterson, Dusan Zrnic, Valery Melnikov, and Alexander Ryhzkov have been
exceedingly helpful over the past several years as we continue our desire to
learn more about how chaff interacts with weather radar systems.

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|  James M. Kurdzo James M. Kurdzo was born in Derby, CT, USA, in 1987. He
received the B.S. degree in meteorology from Millersville University,
Millersville, PA, USA in 2009, as well as the M.S. degree in meteorology, the
M.S. degree in electrical and computer engineering, and the Ph.D. degree in
meteorology from the University of Oklahoma, Norman, Oklahoma, USA, in 2011,
2013, and 2015, respectively. He has been a Technical Staff member at MIT
Lincoln Laboratory since 2016, where he is part of the Air Traffic Control
Systems Group. He has authored over 30 refereed publications and over 100
conference papers/presentations covering a range of research interests,
including radar network and waveform design, severe convective thunderstorms
and tornadoes, phased array weather radar technologies, and weather radar
algorithms. Dr. Kurdzo is a senior member of the Institute of Electrical and
Electronics Engineers (IEEE) and a member of the American Meteorological
Society (AMS). He was a recipient of the MIT Lincoln Laboratory Early Career
Technical Achievement award in 2022, the Tommy C. Craighead award for Best
Paper in Radar Meteorology (Univ. of Oklahoma) in 2015, and several
presentation/paper awards at AMS conferences.
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|  Betty J. Bennett Betty J. Bennett was born in Milwaukee, WI, USA, in 1963.
She received the B.S. degree in computer science from Northern Michigan
University, Marquette, MI, USA, in 1986. Since joining MIT Lincoln Laboratory
in 1988, Betty has worked in a variety of areas with a focus on real-time
systems. She is currently a member of the Air Traffic Control Systems Group at
Lincoln Laboratory working on the NEXRAD Product Enhancement Team. She has
authored or co-authored over 30 refereed articles, technical reports, and
conference papers/presentations.
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|  John Y. N. Cho John Y. N. Cho was born in Tokyo, Japan, in 1963. He
received the B.S. and M.S. degrees in electrical engineering from Stanford
University, Stanford, CA, USA, in 1985 and 1986, respectively, and the Ph.D.
degree in electrical engineering from Cornell University, Ithaca, NY, USA, in
1993. From 1986 to 1988, he was a U.S. Peace Corps Volunteer in Sierra Leone.
From 1993 to 1997, he was a Research Associate at the National Astronomy and
Ionosphere Center’s Arecibo Observatory, Arecibo, PR, USA. In 1996, he joined
the Leibniz Institute for Atmospheric Physics, K$\ddot{\textup{u}}$hlungsborn,
Germany, as a Visiting Scientist. From 1997 to 2002, he was a Research
Scientist with the Department of Earth, Atmospheric, and Planetary Sciences,
Massachusetts Institute of Technology, Cambridge, MA, USA. Since 2002, he has
been with the MIT Lincoln Laboratory, Lexington, MA, USA, where he is
currently a Senior Staff with the Air Traffic Control Systems Group. He has
authored or co-authored over 60 refereed articles and over 60 conference
papers and technical reports. His research interests include atmospheric radar
algorithms, weather sensor benefit quantification, weather effects on air
traffic flow management, atmospheric waves and turbulence, noctilucent clouds,
and meteors. Dr. Cho is a member of the American Geophysical Union and the
American Meteorological Society. He was a recipient of the CEDAR Prize (NSF),
the Young Scientist Award (URSI), and the Best Paper Award (10th USA/Europe
ATM R&D Seminar).
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|  Michael F. Donovan Michael F. Donovan was born in Hudson, MA, USA, in 1963.
He received the B.S. in meteorology and atmospheric science from University of
Massachusetts, Lowell, MA, USA, in 1985. He is currently an Assistant Staff
member at MIT Lincoln Laboratory and has been a member of the Air Traffic
Control Systems Group since 1987. He has authored or co-authored over 10
refereed articles and over 20 conference papers and technical reports. His
research interests include weather radar algorithms, as well as aviation
hazards associated with convective thunderstorms and in-flight icing. He is a
member of the American Meteorological Society.
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