AMTAtmospheric Measurement TechniquesAMTAtmos. Meas. Tech.1867-8548Copernicus PublicationsGöttingen, Germany10.5194/amt-10-2573-2017Retrieval of the raindrop size distribution from polarimetric
radar data using double-moment normalisationRaupachTimothy H.https://orcid.org/0000-0003-3336-7610BerneAlexisalexis.berne@epfl.chEnvironmental Remote Sensing Laboratory, School of
Architecture, Civil, and Environmental Engineering, École
Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne,
SwitzerlandAlexis Berne (alexis.berne@epfl.ch)20July20171072573259416September201625November201623May201710June2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://amt.copernicus.org/articles/10/2573/2017/amt-10-2573-2017.htmlThe full text article is available as a PDF file from https://amt.copernicus.org/articles/10/2573/2017/amt-10-2573-2017.pdf
A new technique for estimating the raindrop size distribution (DSD)
from polarimetric radar data is proposed. Two statistical moments of
the DSD are estimated from polarimetric variables, and the DSD is
reconstructed using a double-moment normalisation. The technique
takes advantage of the relative invariance of the double-moment
normalised DSD. The method was tested using X-band radar data and
networks of disdrometers in three different climatic
regions. Radar-derived estimates of the DSD compare reasonably well
to observations. In the three tested domains, in terms of DSD
moments, rain rate, and characteristic drop diameter, the proposed
method performs similarly to and often better than
a state-of-the-art DSD-retrieval technique. The approach is flexible
because no specific DSD model is prescribed. In addition, a method
is proposed to treat noisy radar data to improve DSD-retrieval
performance with radar measurements.
Introduction
The raindrop size distribution (DSD) describes the microstructure of
liquid precipitation and is highly variable
. The DSD is measured at the point scale by
disdrometers. For applications such as numerical weather prediction
e.g. or radar remote sensing
e.g., it is often necessary to know the areal
DSD at the pixel scale. In other cases, such as studies of the
microphysics of precipitation , it would be useful to be able to remotely infer
the DSD aloft or in remote locations. For these reasons, retrieval of
the DSD from radar data has been a long-standing goal. In this paper
we present a new technique for DSD retrieval from polarimetric radar
data, which is based on the double-moment normalisation technique of
.
Polarimetric weather radars are particularly useful for remote
retrieval of the DSD, because differences between vertically and
horizontally polarised electromagnetic waves reflected off
hydrometeors in the atmosphere provide information on the particles'
concentration, size, and shape. In rainfall, radar reflectivity in
horizontal (ZH, dBZ) or vertical (ZV, dBZ) polarisation
primarily relates to drop concentration and size, differential
reflectivity (ZDR, dB) reflects drop shape, and specific
differential phase shift on propagation (Kdp, ∘km-1) relates to both the concentration and shape of the drops
. showed that ZDR can be
linked to the median volume drop diameter, a microphysical property of
rain. Since then, many methods for DSD retrieval from radar variables
have been proposed.
introduced the “constrained gamma”
method, in which the shape and slope parameters of a gamma DSD model
are assumed dependent. This assumption is
subject to debate e.g.. The technique, modified by
, can provide useful DSD information
. In the “beta” method ,
the effective slope of the drop axis ratio to diameter relationship is
retrieved. The slope is used to find parameter values for the normalised
gamma model of , which has advantages for use with
polarimetric observations . Retrieval of
the gamma model shape parameter with the beta method is subject to high
uncertainty . To deal
with noisy ZDR and Kdp data at low rain rates,
used the beta method for heavy rain
and disdrometer-based regressions on ZH and ZDR for
light rain. found that the constrained gamma method
was in better agreement with disdrometer data than the beta method, while
reported similar performance from the two
techniques, and both studies noted that the beta method is sensitive to
errors in Kdp. developed
a neural-network DSD-retrieval technique, and spatial correlations of DSD
model parameters have been retrieved from radar data .
X-band polarimetric weather radars are popular due to their
portability, small size, and high resolution and sensitivity, but
measurements at X band suffer from attenuation by heavy rain
and must be
corrected . Several
DSD-retrieval algorithms have been developed for X band
e.g., including some with
integrated attenuation correction e.g.. The self-consistent with optimal
parameterization attenuation correction and microphysics
estimation (SCOP-ME) algorithm, developed through studies by
and
, uses relationships calculated for the
Rayleigh limit, corrected for Mie scattering at X band. It performs
well compared to contemporary algorithms and disdrometer observations
. In this paper we present a new method for
DSD retrieval that uses the double-moment DSD normalisation of
, we and compare it to SCOP-ME.
The rest of this manuscript is organised as follows: we briefly
describe the double-moment DSD normalisation technique of
in Sect. . Bulk rainfall
variables that we use are introduced in Sect. . Data used are presented in Sect. . In Sect. we propose a new
DSD-retrieval method that uses double-moment normalisation to retrieve
the DSD from polarimetric radar data. Its performance is compared to
that of SCOP-ME using radar variables simulated from DSD measurements
in Sect. . In Sect.
we introduce a new method to reduce the effects of noise in radar
measurements. Using this method, the DSD-retrieval algorithms are
compared using radar data in Sect. . Conclusions
are made in Sect. .
Double-moment DSD normalisation
The DSD is written as N(D) (mm-1m-3) and is defined as the
concentration in air of raindrops with equivolume diameter in the
interval from D to D+δD mm. The equivolume diameter is used
because raindrops become oblate with size
e.g.; it is simply the diameter of
a sphere that contains the same volume of water as a drop. Mn (mmnm-3), the nth-order moment of the DSD, is
Mn=∫0∞N(D)DndD.
The double-moment normalisation method of allows
for the DSD to be expressed as a combination of two of its moments
Mi and Mj of arbitrary orders i and j and a double-moment
normalised DSD h(x) (–), where x=DMi1/(j-i)Mj-1/(j-i) (–) is the second-normalised diameter . Using the
normalisation, the DSD can be written as
N(D)=Mi(j+1)/(j-i)Mj(i+1)/(i-j)h(x).
The method is flexible because the function h(x) is not
prescribed. suggested that a generalised gamma
model is an appropriate choice for h(x). Following their
recommendation, we use the following double-moment normalised DSD
:
N^(D)=Mi(j+1)/(j-i)Mj(i+1)/(i-j)h^(x),h^(x)=cΓi(j+cμ)/(i-j)Γj(-i-cμ)/(i-j)xcμ-1×exp-ΓiΓjc/(i-j)xc,
where Γ is the gamma function, Γi=Γ(μ+i/c) and Γj=Γ(μ+j/c), and c (–) and μ (–)
are parameters which must be fitted to the generalised gamma
model. Since this formulation allows any DSD to be described using
only two of its statistical moments, the task of our DSD-retrieval
algorithm is to estimate two DSD moments from polarimetric radar data.
The question of whether the double-moment normalised DSD is invariant
has been investigated. Compared to previous single-moment
normalisation approaches that vary by rainfall type
, the double-moment approach shows more
similarity across such changes
. tested the
double-moment normalised DSD across spatial displacement and between
different climatic regions. They showed that for practical purposes in
stratiform rain and with well-chosen input moments, the double-moment
DSD can be considered invariant across space with acceptable resulting
performance on reconstruction of the DSD. showed
that h(x) derived from time series measurements at one location had
low scatter around the average double-moment normalised DSD. In the
DSD-retrieval method proposed here, we make the assumption that the
double-moment normalised DSD function h(x) is invariant in space and
time over the typical domain of interest and that variance in the DSD
is adequately explained through variance in two moments of the DSD.
Bulk rainfall variables
All bulk rainfall variables can be derived from the DSD a detailed review is provided by. The mass-weighted
mean drop diameter Dm (mm), useful as a characteristic drop size,
is M4/M3. Liquid water content W (g m-3) is related to the
third moment of the DSD and is written as
W=π610-3ρwM3,
where ρw (g cm-3) is the density of water. The
rain rate R (mm h-1) is defined as
R=6π10-4∫0∞v(D)D3N(D)dD,
where v(D) (m s-1) is the still-air terminal fall
speed of a drop with equivolume diameter D. In this study v(D) was
calculated using the method of , for
site-specific altitudes and latitudes, and an assumed sea-level
temperature of 15∘ and relative humidity of 0.95.
Radar variables can also be derived from the DSD. In Rayleigh
scattering, when the radar wavelength is much larger than the
particles being measured and drops are assumed to be spherical, the
radar reflectivity is Z=M6. In Mie
scattering, in which the wavelength is of similar size to the
particles, reflectivity in horizontal polarisation Zh (mm6m-3) is defined as Zh=106λ4π5|K|2∫0∞σbh(D)N(D)dD,
where λ (cm) is the wavelength, |K|2 (–) is the
dielectric factor of water, and σbh(D) (cm2) is the
back-scattering cross section at horizontal polarisation of a raindrop
of equivolume diameter D. Reflectivity in vertical polarisation,
Zv (mm6m-3), is obtained by replacing σbh(D)
with the vertically polarised back-scattering cross section
σbv(D) (cm2). It is usual practice to deal with radar
reflectivities in dBZ, calculated as ZH=10log10Zh and ZV=10log10Zv.
Differential reflectivity ZDR (dB) is ZH-ZV. Differential
reflectivity in linear units, ξdr (–), defined as Zh/Zv, has
been shown to relate to the reflectivity-weighted mean drop axis ratio
rz (–) . rz is defined as
rz=∫0∞r(D)D6N(D)dD∫0∞D6N(D)dD,
where r(D) is the vertical to horizontal axis ratio of
a drop of equivolume diameter D. The relationship found by
is
rz∼ξdr-37,
which is valid for narrow distributions of the raindrop axis
ratio .
Dual-polarisation radars measure specific differential phase shift (on
propagation) Kdp (∘km-1), which is the difference in
phase produced between horizontally and vertically polarised waves
that pass through rain. It is defined as Kdp=180λπ10-1∫0∞Refhh(D)-fvv(D)N(D)dD,
where Re represents the real part of a complex number and
Re(fhh) (cm) and Re(fvv) (cm) are the real parts of the
forward scattering amplitudes for horizontal and vertical polarisation
respectively. showed that Kdp can be linked
to the product of liquid water content and the deviation from unity of
the mass-weighted mean raindrop axis ratio rm (–). rm is defined
as
rm=∫0∞r(D)D3N(D)dD∫0∞D3N(D)dD.Kdp can be written as
Kdp=180λ10-1CWρw(1-rm),
with dimensionless value C∼3.75. Various raindrop axis ratio functions are
available e.g.. We return to the question of
axis ratios and Kdp in Sect. .
The integrals in this section and Eq. () are
idealised because the range of drop sizes is written from zero to
infinity. Using measured data, the integrals were calculated over
truncated classes of diameter and second-normalised diameter, with D
and x as class centres and dD and dx as class widths. Since
truncation potentially affects bulk variables
e.g., we used the same truncation limits for
compared quantities. When polarimetric variables were calculated from
DSDs, the T-matrix codes of were used to
calculate raindrop scattering properties, with an assumed temperature
of 12.5 ∘C, a Gaussian distribution of raindrop canting
angles with zero mean and a standard deviation of 6∘stated as reasonable by, and a radar frequency
of 9.4 GHz.
Summary of instrument networks used. Coordinates for
Parsivel networks are bounding boxes. Altitude is above sea level
to nearest 10 m. Hours are provided only for non-instantaneous
measurements and show total hours of rain data measured by each
network.
Data setInstrument typeCoordinatesAltitude (m)HoursHyMeXParsivel (V1 and V2)44.5547–44.6141∘ N, 4.3826–4.5461∘ E200–640403MRR44.5790∘ N, 4.5011∘ E27024X-band radar44.6141∘ N, 4.5461∘ E600PayerneParsivel (V1)46.8115–46.9783∘ N, 6.9184–7.13∘ E430–490347X-band radar46.8133∘ N, 6.9428∘ E490IowaParsivel (V2)41.6406–42.2388∘ N, 92.4637–91.5416∘ W200–290412X-band radar41.8870∘ N, 91.7341∘ W260Data
To train and test the new method, data from three networks of OTT
Parsivel disdrometers were used. Each
network had a nearby X-band weather radar that scanned above the
disdrometers. A full description of the data and their treatment, and
the coordinates for all stations, are provided in
, in which the same disdrometer networks were
used. Here we provide a summary of the data used in this study.
The first network provided the HyMeX data set. This network was located in
Ardèche, France, in the autumns of 2012 and 2013, for the special
observation periods of the Hydrological Cycle in the Mediterranean Experiment
HyMeX
See
http://www.hymex.org.
,. In this study we used
data from 11 first-generation Parsivel and 5
Parsivel2.
disdrometers located in the approximately 13×7km2 network. Also used were data from a METEK GmbH micro
rain radar MRR
within the network, which provided vertical profiles of estimated DSDs
recorded with 100 m vertical resolution and 10 s integration
time. MXPol, a transportable Doppler dual-polarisation weather radar
for instrument details see, was located to the
north-east of the disdrometer network. In 2013, MXPol recorded “stacked”
plan position indicator (PPI) scans above the Parsivel network at elevations
of 4, 5, 6, 8, 10, 12, 14, 16, and 20∘ above
horizontal, with a return time of about 6 min. Six rainfall events in
which the MRR and MXPol both recorded data were selected for 2013. The events
were from 1.8 to 7.5 h in length. Temperature data from a weather
station at Pradel Grainage were used to estimate freezing level cutoff
heights, below which precipitation was assumed to be primarily liquid. These
heights ranged from 971 to 2386 ma.s.l., and only those MRR data
from below the cutoff level per event were used. More network details and the
list of identified events are provided in . The
HyMeX data set was the only set used in which MRR estimates of the DSD aloft
were available.
Instrument stations with corresponding PPI volumes, with
the number of scans for that volume (S), the volume centre's
height above the ground (H, m a.g.l., to nearest 10 m), height
above sea level (H, m a.s.l., to nearest 10 m), and horizontal
range from the radar (D, km). MI (mm h-1) is the maximum
1 min rain intensity recorded by each instrument at a radar
scan time.
Two more data sets were used in order to incorporate data from
different climatologies. The second instrument network was composed of
five first-generation Parsivel disdrometers and MXPol, in Payerne,
Switzerland, and took measurements from February to July 2014. We used
the MXPol PPI scan at 5∘ above horizontal, which had
a return time of about 5 min. The scans covered the region over
three of the disdrometers. The third data set was from a network of 14 Parsivel2 disdrometers deployed in Iowa,
US, during the National Aeronautics and Space
Administration Iowa Flood Studies (IFloodS) Global
Precipitation Measurement ground validation campaign. Overlooking
this network was the University of Iowa's X-band radar XPOL5
. We used PPI data recorded at 3∘
above horizontal, with a variable return time of about 2 to 8 min, for 3 days of heavy rainfall: the 25th, 26th, and 27th of May 2013.
These scans covered the area over 10 of the
disdrometers. The three networks were in regions with different
climatologies as described
in. Table
provides a summary of the three networks. Radar scans were matched to
instrument times by finding scans for which at least half of the
typical scanning period was within the instrument integration
time. Scans typically lasted about 30 s in HyMeX, about 44 s in Payerne, and about 60 s in Iowa. Table shows the instruments covered by PPI scans,
the distance of each station to the PPI radar volumes used, and the
number of radar scans that overlapped with 1 min observations.
Disdrometer data, which had raw integration times of either 30 or 60 s, and MRR data, with 10 s integration time, were resampled to
1 min temporal resolution. HyMeX and Payerne Parsivel data were
corrected with reference to 2D-video-disdrometer (2DVD) measurements
from the HyMeX campaign , with correction factors trained using
reprocessed Parsivel data from the HyMeX 2013 campaign. This procedure
removed unrealistically large drops and those too far from expected
velocities, adjusted velocity measurements, and adjusted drop
concentrations so that DSD moments more closely matched those of the
2DVD. These Parsivel data were quality controlled so that only
error-free time steps containing liquid precipitation were used. Iowa
Parsivel data were used as provided without further quality control.
Parsivel data are subject to uncertainty due to differences across
individual instruments and instrument generations
(e.g.,
),
and their limited sampling area introduces
a bias, as reported by . The Iowa data were
provided in diameter class definitions that differed from those of the
instrument manufacturer . The HyMeX and
Payerne data sets used the manufacturer's diameter class definitions,
which implies the assumption of a raindrop axis ratio to equivolume
diameter relationship . Our tests (not
shown) showed limited differences made to DSD bulk variables when
different axis ratio functions were used to modify the class
definitions. Given the uncertainties involved in using modified
diameter classes, we decided to use the manufacturer's class
definitions for these two data sets. For each of the three regions,
the Parsivel data were randomly sampled so that 60 % of records formed
a training data set and the remaining 40 % formed an independent
validation data set. Sensitivity of the random sampling was evaluated
through repeated tests with different sample realisations and was
found to be low.
All available disdrometer and PPI data were used, while MRR data were
subset to event times so that likely solid precipitation was not
considered. MRR data were attenuation-corrected
and contained DSDs retrieved
with vertical wind ignored . Negative concentrations in
MRR DSDs were reset to zero. PPI radar reflectivities were compared to
measurements from disdrometers (and the MRR in HyMeX), and bias in
ZH was corrected on a per-campaign basis. Bias in ZDR was
estimated using vertical scans birdbath scans, similar
to and was corrected in each of the three
data sets. Two days of radar data from Payerne (22 March 2014 and
8 April 2014) exhibited higher radar bias due to hardware problems and
were not included in this study. Attenuation correction for the PPI
data was performed using the ZPHI algorithm ,
and Kdp was estimated using the method of
. PPI scan data were sampled for instrument
locations by taking the mean values of radar volumes that horizontally overlapped
the instrument coordinates. To discount noise, PPI
records were subset to those for which ZH was greater than or equal
to 10 dBZ, and the signal-to-noise ratio in horizontal polarisation was
greater than or equal to 5 dB. DSD data were treated as in
: Parsivel DSDs were truncated to 0.2495
(0.2565) to 7 (7.21) mm for HyMeX and Payerne (Iowa) Parsivel data
; to avoid including overestimated numbers of
small drops , DSDs estimated by the MRR were
truncated to 0.6 to 5.8 mm and MRR data
were further subset to records with R≤150mmh-1 (thus
removing 0.2 % of records); MRR data for altitudes greater than 2250 m
were excluded because not enough points were available at those
altitudes, and all DSDs were subset to time steps in which R>0.1mmh-1. In each data set, more than 85 % of the DSDs sampled were
classified as stratiform type by .
To compare measured vs. estimated or retrieved values in this work,
we use the median relative bias, the interquartile range (IQR) of
relative bias, and the squared Pearson correlation coefficient (r2)
between reference and estimated values. If VR is the reference
value and VE is the estimated value, the relative bias expressed as
a percentage of the reference value is defined as 100(VE-VR)/VR.
DSD retrieval from polarimetric radar data
showed that with reasonably chosen input
moments, the double-moment normalised DSD of can
be assumed invariant across spatial displacement in stratiform rain,
with a performance loss that is acceptable for practical
applications. Results on limited data for non-stratiform rain types
suggested that, while the double-moment normalised DSD varies more in
these cases, the assumption of its invariance may still lead to
acceptable performance with input moments that are not both of low or
both of high order. Using the assumption of an invariant double-moment
normalised DSD model, the DSD can be estimated using polarimetric
radar data. Given a known double-moment normalised DSD, the task of
DSD reconstruction becomes that of estimating from radar information
the values of two DSD moments. In this section we present a new
DSD-retrieval method that uses this idea. The aim of the proposed
DSD-retrieval technique is to retrieve two DSD moments using only
polarimetric radar data.
The SCOP-ME method was trained with DSDs simulated using a DSD model
and a wide range of DSD parameter values. In contrast, we used
empirical DSDs measured by Parsivels to train our method to avoid any
assumption about the shape of the DSD. A trade-off in using
empirical DSDs is that measured DSDs are necessarily truncated by
instrumental limitations.
However, previous studies
have shown that if the considered range of drop diameters is large
enough around the median drop diameter D0 (mm), the effect of
truncation on calculated bulk variables is limited
.
concluded that the effect of maximum considered drop size
Dmax on bulk variables is negligible if
Dmax exceeds 2.5D0. Using D0 calculated from the
recorded (truncated) Parsivel DSDs, this criteria was met for 99.6 %
of the records. The criteria of is that,
for there to be less than five percent error on bulk variables, the
minimum drop size Dmin should be less than D0/2 and
Dmax should exceed 4D0. This constraint was met by
90.5 % of the DSDs (93.5 % met this criteria for the upper drop size
limit). Calculated D0 may also be subject to error because of the
truncation, but we consider that these calculations give broad
confidence in the bulk variables we used to train the method. Further,
the truncation on the Parsivel data affects primarily very small drops
since large drops are rare, and therefore its influence on the
higher-order moments we use is expected to be negligible.
The training data set was sampled as 60 % of each of the three
Parsivel data sets and contained 181 829 measured 1 min
DSDs. ZH, Kdp, and ZDR were calculated for these DSDs for the
MXPol stacked PPI incidence angles; temperatures of five, 10, and 15 ∘C; and each of four drop axis ratio functions: those of
, and that of in the form
shown in . Unusual records with ZDR
or Kdp less than or equal to zero (0.16 % of all simulated radar
records) were excluded.
Retrieval of DSD moment six
Radar reflectivity in linear units, Zh (mm6m-3), is the
sixth moment of the DSD in the Rayleigh scattering regime for
spherical drops . At X-band frequencies, larger
drops enter into the Mie scattering regime and differences appear
between M6 and Zh. We use the observation that Zh departs
from M6 for heavier rain and assume that this departure occurs
when ZH is greater than a threshold value. This threshold was
determined through comparison of M6 and Zh for DSDs, classed by
ZH in classes of width 2 dBZ between 10 and 40 dBZ, and was set
to 28 dBZ. For both smaller and larger reflectivity values, a power
law relationship was found using orthogonal least squares fitting in
log–log space. The resulting relationship is
M6^=Zh1.01if 10log10(Zh)≤282.67Zh0.86if 10log10(Zh)>28.
On the training set, median relative bias between M6^ and
M6 was 0.1%, the IQR of relative bias was 2.5 percentage points,
and the r2 value was 0.98. The fitted relationship is shown on
samples of training data in Fig. . Temperature
made only limited difference to the fitted parameters: the pre-factor
varied from 2.45 to 2.90 for the larger values of ZH, and the other
parameters differed by 0.01 or less from the value found for all
temperatures combined.
A sample of 20 000 points from the training set, showing
the relationship between radar reflectivity and DSD moment six in
dB scale. The one-to-one line is shown in black; the red dashed
line shows the fitted relationship of Eq. (). The ZH threshold of 28 dBZ is shown with
a triangle.
Retrieval of DSD moment three
Retrieving a second, lower-order DSD moment is more difficult than
estimating M6, because radar variables are more closely linked to
the higher-order moments of the DSD. Using theoretical relationships
as much as possible, we present a method to estimate the third moment
of the DSD from polarimetric data. As shown in Eq. (),
the reflectivity-weighted mean drop axis ratio, rz, is related to
a negative power of the differential reflectivity in linear units. In
, the reflectivity-weighted and
mass-weighted drop axis ratios were assumed to be the same and
differences were dealt with through fitting of qualitative
relationships between radar variables. A similar approach is taken
here. Since rz and the mass-weighted mean drop axis ratio rm are
both weighted mean drop axis ratios, we assume that rm is also
related to differential reflectivity and estimate rm using
a polynomial fit to ZDR, such that
rm^=∑i=05ciZDRi.
With our training data, this polynomial of order five produced low
relative bias on retrieval of M3. Recall from Eq. ()
that M3 relates to W: substituting Eq. () into
Eq. (), and solving for M3, we have
M3=6λ10318πCKdp(1-rm).
At X band (9.4 GHz, λ=3.189cm), assuming that
ρw=1gcm-3, and replacing rm with its estimate based
on ZDR, M3 is predicted by
M3=338.4C^Kdp(1-rm^),
where C^ is a single representative value for
C.
Kdp is sensitive to the raindrop axis ratio
e.g., so values for ci and C^
were found per axis ratio function. The coefficients ci in Eq. () were found using least squares polynomial
fitting. In rare cases for large values of ZDR the relationships
returned unrealistic values of rm (0≤rm or rm>1). In
these few cases, rm^ was set to 0.75. Estimated
rm^ values were used to find C for each training DSD,
and the mean of these values was used as C^. The results
and their performance statistics are shown in Table . Fitted parameters differed across the three
tested temperatures. However, parameters fitted using all training
data performed similarly on training data for individual temperatures,
with the median relative bias remaining within ±1% of the
all-temperatures value and IQR of relative bias varying by about one
percentage point. The values fitted using combined training data were
used.
Fitted values of C^ (Eq. ) and ci (Eq. ), by
drop axis ratio function (Ratio). M3 estimation performance in
the training data is shown in terms of median relative bias (RB, %), IQR of relative bias (IQR, % pts), and r2. Max ZDR
(dB) shows the maximum value of ZDR each relationship can use.
RatioC^c0c1c2c3c4c5RBIQRr2Max ZDRThurai3.4561-0.0736240.041651-0.0170420.002498-0.0000930.8250.976.58Brandes3.3111-0.0776720.047704-0.0200420.003505-0.000220-0.7220.978.51Andsager3.2561-0.0901370.070235-0.0339330.006913-0.000514-0.2200.977.15Beard3.2171-0.0876460.053086-0.0203360.002963-0.000129-0.6210.977.21Summary of DSD-retrieval technique
The proposed DSD-retrieval method is summarised as follows: the
double-moment normalised DSD h^(x) with parameters c and
μ is assumed trained from data and known. Then, given Kdp,
ZDR, and Zh, (1) DSD moment six is estimated using Eq. () and (2) DSD moment three is estimated using
Eqs. () and () and parameters
from Table . The DSD is then retrieved using
Eqs. () and ()
with i=3 and j=6.
Comparison to an existing DSD-retrieval method
The new DSD-retrieval method was compared to SCOP-ME
. We implemented SCOP-ME using its
description in . SCOP-ME was developed for
X band using simulated DSDs and T-matrix simulations of radar
variables, and in it is shown to outperform
the algorithms of and
. The DSD model used by SCOP-ME is based on the
normalised DSD of see
also. also shows
SCOP-ME equations in which Kdp is not used, which are to be
employed when Kdp is absent or close to noise. In this work we used
only the version of SCOP-ME that uses Kdp, as presented in
, and we use it only with positive and
non-zero values of Kdp and ZDR (tests on our data sets showed
better SCOP-ME performance with this configuration).
provides an explicit expression for
rain rate using polarimetric variables, but, since we are interested in
the whole DSD, in the following we compare R computed from
reconstructed DSDs. The comparison of the two methods is first shown
using Parsivel data in which the radar values were simulated using
T-matrix codes and were therefore free of radar measurement noise.
Comparisons of the two techniques were made using the Parsivel
validation data set composed of 40 % of the records from HyMeX,
Payerne, and Iowa. For each 1 min DSD, Zh, Kdp, and ZDR
were calculated using T-matrix codes, for an elevation angle of
4∘ above horizontal, and using each of the four drop axis
ratio functions. For the double-moment technique, the generalised
gamma model h^(x) (Eq. ) for
i=3 and j=6 was used. h^(x) was fitted to non-zero median
values of h(x) in classes of x with width 0.2, using weighted
least squares fitting in log space, with each class weighted by
nC4, where nC is the number of observations in the class. This
is the same technique shown in , except that
the class weights increase more quickly with the number of class
points. We found this different weighting improved the accuracy of the
shape of the average reconstructed DSD, which is sensitive to the form
of h^(x). The parameters found for the combined Parsivel
training data were c=1.69 and μ=2.22. SCOP-ME and the
double-moment method were used to retrieve the DSD concentrations
N(D) for D in the class centres of the truncated Parsivel diameter
classes. For each technique and axis ratio function, retrieved DSDs
were compared to measured DSDs by comparing moments zero to seven,
Dm, and R.
Comparisons of relative error distributions by technique are shown in
Fig. . Example scatter plot results are
shown for the HyMeX data set and the drop axis ratio model of
in Fig. . The
Beard model, which has been shown to match well to observations
, is shown because it provided the equilibrium
drop shapes around which the SCOP-ME training set was simulated
. Because we are using empirical DSDs,
some extreme values of Dm are shown in this figure. Values of Dm
above 5 mm are extremely rare; less than 0.02 % of DSDs in each data
set show these values, and they have a negligible influence on the
regression lines. Full performance results are shown for the HyMeX
data set in Table , for Payerne in Table , and for Iowa in Table . The metrics used were median relative bias,
IQR of relative bias, r2, and the slope of the linear regression on
measured vs. reconstructed points. Differences between the performance
metrics for the two techniques were calculated such that a negative
difference indicates that the double-moment technique performed better
than SCOP-ME. These differences are shown visually in Fig. , in which red colours show
negative differences.
Relative bias distributions for the double-moment and SCOP-ME
DSD-retrieval methods, by drop axis ratio function and data set (H
stands for HyMeX, P for Payerne, and I for Iowa). Variables are
moment order n (mmnm-3), Dm (mm), and R (mm h-1). Bold bars show medians, boxes show IQRs, and whiskers show
10th to 90th percentile ranges.
Density scatter plots of retrieved vs. measured moments
Mn (mmnm-3), R (mm h-1), and Dm (mm) for the
double-moment method, on the HyMeX data set, using the axis ratio
function of . One-to-one lines are shown in
black. Regression lines for the double-moment method are shown in
solid red, and dotted red lines show linear regressions for SCOP-ME,
for which the densities are not shown.
Differences in performance between the double-moment
technique and SCOP-ME, using radar variables simulated from Parsivel
data, by region and drop axis ratio function (differences in Tables , , and
). Reds indicate negative differences, where
the double-moment technique outperformed SCOP-ME. Variables are
moment order n (mmnm-3), Dm (mm), and R (mm h-1). Differences are shown for median relative bias (RB, % pts), IQR of relative bias (IQR, % pts), r2 (difference in
deviations from unity, multiplied by 100 for display on this scale),
and regression slope (S; difference in deviations from unity,
multiplied by 100).
Average differences between double-moment and SCOP-ME
techniques, on Parsivel data, over three regions and four
axis ratio functions. Negative values show an improvement by the
double-moment technique over SCOP-ME.
In over half of the tested region, axis ratio function, and variable
combinations, the double-moment technique produced a better median
relative bias than the SCOP-ME technique, with an overall average
difference of -0.53 percentage points. IQR of relative bias was
usually slightly higher for the double-moment technique, with an
average difference of 3.8 percentage points. Correlation coefficients
and scatter plot slopes were usually similar for both techniques. The
average differences across the three tested regions and four tested
raindrop axis ratio functions are shown in Table . On average, the double-moment technique
produced better median relative bias than SCOP-ME on R, Dm, and
DSD moments two to seven. IQRs were similar on average, with the
exception of moments zero, one, and seven for which SCOP-ME produced
notably smaller IQRs. As shown in Tables ,
, and , the results
differed across the different drop axis ratio functions and
regions. It was often the case that SCOP-ME produced a less biased
estimate of DSD moment zero, but in most of these cases the
double-moment technique produced a better r2. The double-moment
technique's performance variations relate to the accuracy of the
prediction of DSD moment three from Kdp and ZDR and to the fit
of the generalised gamma function h^(x). h^(x) was
trained on data from all data sets combined, in order to have the most
general model possible. Our experiments showed that performance for
low-order moments could be increased in any one region by training the
gamma model on data from that region only. This aligns with the
conclusions of , who noted that, while the
double-normalised DSD can be assumed invariant for practical purposes,
some residual variability remains and results in performance loss that
depends on the input moments used. We now move to testing the two
techniques on measured radar data, in which noise is a problem that
must be dealt with.
Reducing the effects of noise
Radar data are noisy at light rain rates, particularly for Kdp and
ZDRe.g.. We
propose here a method to deal with this noise for the current
application of DSD retrieval. Regressions on Zh and ξdr are
used to determine “expected” values for these variables, which can
be used when the measured values are likely to be noisy. We found that
ZDR can be reasonably predicted from Zh using
Z^DR∼αZZhβZ,
and Kdp can be predicted from Zh and ξdr using
K^dp∼αKZhβK1ξdrβK2,
with parameters αZ, βZ, αK,
βK1, and βK2. Least squares fitting in log–log space,
using the training data set described in Sect. , was used to find best-fitting parameter values
per raindrop axis ratio function. Just as for the retrieval of DSD
moment six, assumed air temperature made only a small difference
(parameter values fitted to individual temperature data sets differed
by less than 6 % from those fitted using combined temperatures),
whereas different axis ratios produced more diverse parameter
values. Resulting parameter values and performance statistics are
shown in Table .
Fitted coefficients and the performance of the fits on the
training data, for Eqs. () and
(), by raindrop axis ratio function
(Ratio). Performance is shown in terms of median relative bias (RB, %) and the IQR (% pts) of relative bias.
Threshold values are used to determine when Kdp and ZDR may be
noisy. A threshold value on ZH selects values of ZH for which
Kdp and ZDR showed large variation around their expected values
in the three radar data sets used here. Threshold values on ZDR and
Kdp are those of . Although these were established for
S band, their influence is rather limited compared to the threshold on
ZH. The threshold on ZH is
set to 37 dBZ; it is used in addition to the noise thresholds on
ZDR and Kdp in order to avoid effects like ground clutter as
well as noise that significantly affect any DSD-retrieval algorithm.
A drawback of this method is that it reduces the benefit of observed
polarimetric variables when ZH is under 37 dBZ, but the average
error is kept low.
To reduce the effects of noise, then, if ZH<37 dBZ or ZDR<0.2dB, measured ZDR is replaced by the expected value
Z^DR and ξdr is replaced by
10(Z^DR/10). Likewise, if ZH<37 dBZ or
Kdp<0.3∘km-1, Kdp is replaced by
K^dp (calculated with ξdr^ if
ξdr was replaced). This treatment method allows radar data with
negative or zero Kdp or ZDR to be used. The treatment improved
DSD-retrieval performance for both the double-moment and SCOP-ME
techniques. For example, on PPI data with positive ZDR and Kdp,
when retrieved DSDs were matched to measured MRR data (Sect. ), the median relative bias was reduced by an average
(across variables) of ∼8 percentage points for SCOP-ME and by
∼16 percentage points for the double-moment technique, while
average IQRs were reduced more; for example on the comparison with MRR
data the IQRs were reduced by ∼78 (80) percentage points for the
SCOP-ME (double-moment) method. When retrieved DSDs were compared to
Parsivel data (Sect. ), the noise in the radar
data contributed to errors to such an extent that for both techniques
the proposed treatment reduced the IQR and at times the median of
relative bias by hundreds of percentage points for some variables. We
note that because most values of ZH recorded in the PPIs analysed
here were lower than 37 dBZ, the noise correction affected the
majority of radar records.
Comparisons using radar data
The DSD-retrieval techniques were applied to PPI radar data from the
three locations. The double-moment technique was run on
noise-corrected data. SCOP-ME was run on uncorrected PPI data (subset
to Kdp>0 and ZDR>0) and noise-corrected data. We used
the elevation angles of the stacked PPIs for HyMeX, 5∘ for
Payerne, and 3∘ for Iowa. Measured radar variables ZH,
Kdp, and ZDR were recovered for volumes corresponding to
instrument locations. DSD retrieval was performed using these values,
and the resulting DSDs compared to those that were measured by other
instruments. All comparisons using PPI data involved a difference in
measurement volume – a change-of-support problem that we expect will
introduce error spread e.g.. There were,
at times, significant vertical distances between the radar volume and
the ground-based Parsivels used in these comparisons (see Table ). Further, the disdrometers and MRR made
integrated measurements, while the PPI scans made instantaneous
measurements that could shift in time compared to the instrument's
integration period (at worst, this shift could lead to the scan being
up to half the typical scan length before or after the
instrument integration time). We used the lowest possible radar
elevation angles, but the possibility remains that at large distances
the radar could have sampled solid precipitation above the ground
instruments. These factors and uncertainty in the noise-correction and
bias-correction techniques combine to create greater uncertainty in the comparisons of
the two techniques made using real data than in those made using
simulated radar variables from disdrometer data.
Because the axis ratio of produced good
results using the double-moment technique on the Parsivel data, the
double-moment technique was used with parameters for this axis ratio
function. Note that the assumption of axis ratio function affects only
parameters of the double-moment technique, because the radar data used
in this section are measured, not simulated, and the SCOP-ME technique
is used as presented in . In the HyMeX
campaign, the lowest available PPI elevation angle (4∘) was
used to compare results to Parsivels, but there was also an MRR at
Pradel Grainage which retrieved estimates of the DSD
aloft. MRR-derived DSDs were compared at eight different altitudes
using the MXPol stacked PPIs (except 20∘ elevation) above the
HyMeX instrument network. We first address the comparisons with MRR
for HyMeX and then move to the comparisons with the Parsivel networks in
all three regions.
Distributions of relative bias on DSD moments zero to
seven, comparing DSDs retrieved using PPI data to those measured
by the MRR at Pradel Grainage. The results are classed by altitude
for a selection of three altitudes across the compared range. Two
comparisons are shown: in comparison A, SCOP-ME used raw PPI data
and the double-moment technique used the noise-corrected version
of the same data set. In B, both techniques used noise-corrected
data sets. Variables and symbols as for Fig. . Input data for comparison A excluded
records with negative or zero Kdp or ZDR, while the
noise-correction technique deals with these values and they were
included in comparison B.
Distributions of relative bias on DSD moments zero to seven,
comparing DSDs retrieved using noise-corrected PPI data and those
measured by Parsivel networks. Variables and symbols as for Fig. .
Comparisons to MRR DSD estimates aloft
MXPol volume centre altitudes were projected onto MRR altitude classes
for comparison. The double-moment DSD-retrieval algorithm was used
with generalised gamma model h^ parameters (Eq. ) for MRR data and i=3 and j=6. These
parameters were found using the same fitting technique as for Parsivel
data (Sect. ) but differ since instrumental
differences produce different forms of h^(x). The parameters were set to c=1 and μ=5.28. The use of stronger weighting in the fitting procedure
reduced the influence of the large numbers of small drops returned by
the DSD-retrieval algorithm used by the MRR, and thus the value of
μ was smaller than those reported in . The
reconstructed DSDs were found for classes of drop diameter with class
centres from 0.65 to 5.75 mm and a class width of 0.1 mm, so the
reconstructed truncation matched that of the MRR data. PPI values from
eight 100 m altitude classes between about 900 and 2100 ma.s.l. were compared to MRR estimates of the DSD aloft. Two output
pairings are shown here: the first in which both techniques used
noise-corrected data and the second in which the SCOP-ME technique
used raw data and the double-moment technique used the same raw data
set corrected for noise. This second pairing was made to ensure that
the performance of SCOP-ME was not compromised by the noise-correction
technique.
Results of comparisons between MRR- and PPI-derived DSDs are shown for
three example altitudes in Fig. . There
was good agreement between the recorded radar reflectivity recorded by
both instruments, with a median relative bias of -3%, an IQR on
relative bias of 16 percentage points, and a value of r2 of
0.68. The improvement in SCOP-ME performance made by the noise
correction is clear. When both techniques used noise-corrected input,
both overestimated DSD moment orders zero to four and underestimated
orders six and seven. Rain rate was recovered with a median relative
bias of 8 % (IQR 101 % pts) by the double-moment technique and 17 %
(IQR 105 % pts) by SCOP-ME. The double-moment technique showed lower
median relative bias than SCOP-ME on moments one to five, seven, and
R and smaller IQRs on moments two to seven and R. Similar to some
of the Parsivel results, the double-moment technique overestimated
moments zero and one of the DSD. r2 values were low for both
techniques (the maximum was 0.35, by SCOP-ME for Dm), but the
double-moment technique had the same or a slightly higher value of
r2 in the majority of cases. High best-fit slopes were observed for
both techniques for moments five, six, and seven and show the effect
of a few outlier points in these cases. Performance differences
between the two techniques using noise-corrected data are shown in
Table . Overall, the double-moment technique
for DSD-retrieval out-performed SCOP-ME for the retrieval of DSD
moments above order zero and rain rate measured aloft by the MRR.
Comparisons to DSDs measured by Parsivels
DSDs retrieved from polarimetric radar data were also compared to
those recorded by ground-based Parsivels in the three regions we
studied. Unlike in previous sections where training and validation
divisions of the Parsivel data were used, here we compared DSDs
derived using independent radar data to all available matching
Parsivel records. The DSDs were retrieved in truncated Parsivel drop
diameter classes, using the Parsivel generalised gamma model
parameters quoted in Sect. . Particularly in
the Payerne data set, the noise-correction routine was required in
order to retrieve realistic DSDs; the results shown here are thus for
the SCOP-ME and double-moment techniques both run on noise-corrected
PPI data. Figure shows distributions of
DSD-retrieval relative error for each region.
Differences in performance between the double-moment
technique and SCOP-ME using noise-corrected radar data, for MRR and
for Parsivels by region (differences in Table ). Variables and performance statistics as
for Fig. . Red indicates that
the double-moment technique outperformed SCOP-ME. Grey indicates an
r2 difference greater than 50 on this scale; these slopes were
affected by outliers.
The double-moment technique produced smaller IQRs of relative bias
than SCOP-ME for moments four to seven. For moment orders zero to
three, the double-moment technique produced better median relative
bias than SCOP-ME in the HyMeX and Iowa data sets, but worse in
Payerne. Where the double-moment technique produced better median
relative bias, the average improvement was of seven percentage points;
in cases where SCOP-ME performed better, the average improvement was
five percentage points. Values of r2 and scatter plot slope were
similar between the two techniques, with the majority of cases showing
differences of less than 0.05 for both variables. Differences in
performance between the two techniques are shown in Fig. and Table .
The performance of the double-moment technique is reliant on how
accurately two DSD moments can be extracted from radar data and in
turn on how accurate the radar data are. Both retrieval techniques
appear to be similarly affected by radar inaccuracies, and experiments with different reflectivity bias corrections
(not shown here) showed similar patterns of results. In Parsivel
comparisons, the proposed DSD-retrieval technique was applied using
a single double-moment normalised DSD model in all three tested regions,
without significant performance loss between regions. This supports
previous findings that for practical use
with real radar data in primarily stratiform rain, the double-moment
normalised DSD may be considered invariant in regions at similar
latitudes.
Conclusions
Given the assumption of an invariant normalised DSD, and an estimate
of that function, the DSD can be predicted using only two of its
moments using the double-moment normalisation method of
. Two DSD moments are available from polarimetric
radar data. At X band, radar reflectivity can be used to accurately
predict the sixth moment of the DSD, and moment three can be retrieved
relatively accurately using Kdp and ZDR. We showed that by
estimating these two DSD moments from radar data, the DSD for a radar
volume can be predicted using the double-moment formulation. Tests on
disdrometer data from three networks in different climatic regions
showed that DSD retrieval using this new technique produced similar or
slightly better performance than the SCOP-ME DSD-retrieval technique
of . The proposed method is also more
flexible, because there is no prescribed functional form for the
double-moment normalised DSD, and even a non-parametric h^(x)
could be used. The shape of the retrieved DSD is sensitive to the
fitted form of h^(x), and better fitting methods may in future
improve the performance of the retrieval method. The flexibility of
the technique extends to there being no prescribed method for DSD
moment extraction, which means that the moments used could be tailored
to the intended purpose.
A new method for treatment of radar data with possibly noisy values of
Kdp and ZDR was proposed. The method is based on predicting the
expected values of these variables from radar reflectivity, and it
considerably improved the performance of both the DSD-retrieval
techniques when real radar data were used. DSDs were predicted from
polarimetric variables in noise-corrected PPI scans measured by X-band
radars in each of the three regions. Comparisons of the retrieved DSDs
to MRR data for DSDs aloft in the HyMeX region in France, and of
radar-retrieved DSDs to disdrometer data from the three regions,
showed reasonable agreement but large error spread for both
methods. This study provides a proof of concept for DSD retrieval
using noise-corrected radar data, the double-moment normalisation
method of , and a generalised gamma model for the
normalised DSD. Performance improvements may be possible through
future work that should test the approach using different instruments
and data sets, including testing whether such extensive noise
correction is always required. Future work should also address more
precise prediction of low-order DSD moments from polarimetric radar
data and investigate different models and fitting methods for the
double-moment normalised DSD.
HyMeX MXPol, MRR, and Parsivel data are available through
the HyMeX database at http://www.hymex.org, and DOIs for individual data sets
are listed in Nord et al. (2017). Weather station data, Payerne radar data, and
Payerne Parsivel data are publicly available upon request from EPFL-LTE.
IFloodS X-band radar data is publicly available in Krajewski et al. (2016).
IFloods Parsivel data is publicly available in Petersen et al. (2014).
Comparison of double-moment method to SCOP-ME results on
all Parsivel data in the HyMeX data set by axis ratio function
(Ratio). RB (%) is median relative bias, IQR (% pts) is
interquartile range of relative bias (% points), r2 is squared
correlation coefficient. S is the slope on measured
vs. reconstructed regression. Difference is the difference in absolute
values for RB and IQR and the difference in deviation from unity for
r2 and slope. A negative difference shows that the
double-moment method improved on SCOP-ME's performance.
Comparison of double-moment method to SCOP-ME results on
all Parsivel data in the Payerne data set by axis ratio function
(Ratio). Columns are as for Table .
Comparison of double-moment method to SCOP-ME results on
all Parsivel data in the Iowa data set by axis ratio function
(Ratio). Columns are as for Table .
Differences in performance by variable and region, for DSDs
retrieved from noise-corrected PPI data using the double-moment
technique and SCOP-ME, compared to the MRR at Pradel Grainage
(MRR) and Parsivels by region (HyMeX, Payerne, and Iowa). Metrics
and differences are defined as for Table . An exception is Z, which refers to
ZH measured by the radar, not reconstructed through
DSD retrieval (hence it is the same for both techniques).
The authors declare that they have no conflict of
interest.
Acknowledgements
For deploying and maintaining the instruments used, we thank, for
HyMeX, Joël Jaffrain, Marc Schleiss, Jacopo Grazioli, Daniel Wolfensberger,
André Studzinski (EPFL LTE), Brice Boudevillain, Gilles Molinié,
Simon Gérard (Laboratoire d'étude des Transferts en Hydrologie et
Environnement, LTHE, Grenoble University), Yves Pointin, and Joël Van
Baelen (Laboratoire de Météorologie Physique, LaMP,
Université Blaise Pascal de Clermont-Ferrand). For Payerne, we thank
Joël Jaffrain and Meteosuisse. We thank Jacopo Grazioli for processing the
MRR and radar data sets. HyMeX data were obtained from the HyMeX
program, sponsored by grants MISTRALS/HyMeX, ANR-2011-BS56-027
FLOODSCALE project, OHMCV, LaMP, and EPFL-LTE. We thank the Swiss National
Science Foundation for financial support under grant
2000021_140669. We thank Leo Pio D'Adderio and one anonymous referee for their
constructive reviews.
Edited by: Saverio Mori
Reviewed by: Leo Pio D'Adderio and one anonymous referee
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