Analytically Derived Relations for Rain Estimation Using Polarimetric Radar Measurements
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Analytically Derived Relations for Rain Estimation
Using Polarimetric Radar Measurements
J. Vivekanandan*, Guifu Zhang and Edward Brandes
National Center for Atmospheric Research
P. O. Box 3000
Boulder, CO 80307-3000
Abstract
parameters, and derived a µ-Λ relation from the video-
Polarimetric radar has been successful in disdrometer measurements collected during a special
characterizing cloud and precipitation. Polarization field experiment in east-central Florida evaluating the
parameters such as radar reflectivity (Z), differential potential for polarimetric radar to estimate rainfall in a
reflectivity (ZDR), linear reflectivity difference (ZDP), subtropical environment.
specific differential phase shift (KDP), linear de-
polarization ratio (LDR) as well as the correlation 1. Introduction
coefficient (ρHV) have been successfully measured.
Polarimetric measurements provide more information The accuracy of rain rate estimation by well-calibrated
about precipitation and allow better characterization of radar is limited by the lack of detail knowledge of drop
hydrometeors, accurate rain rate estimation, and size distribution (DSD). Rain rate is usually estimated
retrieval of rain drop size distribution (DSD). In the from radar reflectivity using a Z-R relation based on
past, rain rate (R) estimation from S-Pol was based on convective or stratiform rain. The Z-R relation was
empirical models such as Z-R, R (Z, ZDR) and R (KDP) obtained by fitting gauge measurements and radar
relations, which were derived from regression analyses reflectivity. It is known that the Z-R relation changes
of radar and rain gauge measurements or numerical from location to location and time to time depending
simulations. These fixed empirical relations, however, on changes in DSD. Therefore a fixed empirical Z-R
cannot give accurate estimation results for various relation cannot provide accurate rain estimation for
types of rain and are very sensitive to selection of various types of rain because it cannot handle variation
dataset for fitting, threshold, sampling effects, in drop size information. The relation between radar
measurement errors and so forth. reflectivity and rain rate is almost completely
quantified only if the drop size distribution is
Accurate rain rate estimation requires understanding of specified because they are proportional to moments of
rain microphysics and knowledge of raindrop size DSD; namely, reflectivity is the 6th moment and rain
distribution (DSD), shape and canting angle, while the rate is proportional to the 3.67th moment of the drop
information about rain DSD is important in spectrum. Accurate rain rate estimation requires
understanding rain development and evolution. In the detailed knowledge of rain DSD and hence various
past, rain DSD was commonly assumed to be an rain rate estimators are derived using polarimetric
exponential distribution (this may be true for a long radar observation that includes reflectivity, differential
sample period) but recent observations indicate that reflectivity and propagation phase [Doviak and Zrnic,
the instant rain DSD is better characterized by a 1993].
Gamma distribution. The three Gamma DSD
parameters are usually difficult to retrieve. During the The polarimetric radar technique has attracted great
analysis of data measured by a video-disdrometer, we attention because most of the hydrometeors are non-
found there is a high correlation between the shape (µ) spherical. Particularly in the case of raindrops there is
and slope (Λ) a well-defined relation between size and shape [Seliga
and Bringi, 1976; Oguchi, 1983; Vivekanandan,
_________________________________________ 1999a]. The idea of using differential reflectivity in
rain estimation was first proposed by Seliga and
* Corresponding author address: Bringi in 1976. Much progress has been made in
J. Vivekanandan implementation of a polarization radar measurement
NCAR/RAP, P. O. Box 3000 system and a microphysical retrieval algorithm [Bringi
Boulder, CO 80307 – 3000 and Hendry, 1990]. Polarization parameters such as
USA radar reflectivity (ZHH), differential reflectivity (ZDR),
Email: vivek@ucar.edu linear reflectivity difference (ZDP), specific differentialphase shift (KDP), linear de-polarization ratio (LDR), distribution with µ = 0. It has been found that the three
as well as the correlation coefficient (ρHV) have been parameters are not mutually independent [Ulbrich,
successfully measured. These multi-parameter 1983; Haddad et al. 1997]. Haddad et al (1997)
measurements provide additional information about parameterized rain DSD with transformed parameters
precipitation and allow better microphysical which are uniformly random. The effort has
characterization of hydrometeors. In general, ZHH, concentrated on generating Gamma DSD from
ZDR, and KDP are used for estimating rain rate and drop independent random variables.
spectrum, since they depend mainly on drop size and
shape [Jameson, 1983a&b]. LDR and the covariances Since the three DSD parameters do not correspond to
are used for retrieving canting angles because they are physical parameters such as liquid water content or
sensitive to particle orientation [Ryzhkov et al., 1999; median volume diameter, various normalization
Vivekanandan et al., 1999b]. techniques were used [Willis, 1984; Dou et al., 1999].
Since the dimension of N0 is ill-defined, Chandrasekar
Usefulness of the fixed power-law rain rate and Bringi (1987) proposed to use the total number
estimators is limited by various factors. Although it is concentration Nt instead of N0. Furthermore, a
well known that the polarimetric radar measurements normalized Gamma distribution was first proposed by
contain information regarding rain DSD, the study of Willis and recently adopted by Illingworth and
retrieving rain DSD from polarimetric radar Blackman to eliminate the dependence between N0
measurements is limited to retrieving all three and µ [Willis, 1984; Illingworth and Blackman, 1999;
parameters of a Gamma function [Seliga and Bringi, Testud, 2001]. They recommended using physically
1978; Richter and Hagen, 1997]. An estimate of KDP meaningful parameters to characterize a Gamma DSD.
involves the averaging of differential phase (φDP) over Nevertheless, the number of parameters is the same,
range, hence R(KDP) may overestimate or and the DSD expression becomes more complicated.
underestimate rain rate due to the range averaging of In practice, there is no simplification of the DSD
φDP [Doviak and Zrnic, 1993]. R(ZHH, ZDR) may give function except that DSD parameters are expressed
acceptable rain estimation in some cases, but does not using NT, LWC, and D0.
provide rain drop size distribution (DSD).
In this paper, we study constrained Gamma rain DSD
For a standard sampling time such as 1 minute, and its application to rain estimation from polarimetric
however, some observations indicate that natural rain radar measurements. The paper is organized as
DSD contains fewer of both very large and very small follows: In section 2, video disdrometer measurements
drops than an exponential distribution [Ulbrich, 1983; are analyzed. The three parameters of Gamma DSD
Tokay and Short, 1996]. Estimated rain rate may be are obtained from three moments of the measured
comparable to the actual rain rate of the measured DSD. A constrained relation between µ and Λ is
spectrum using either exponential distribution or derived from disdrometer observations. In section 3,
Gamma distribution when the 3rd or 4th moment is we derive closed-form expressions for calculating rain
included, because rain rate is proportional to the 3.67th rate and characteristic size of raindrops. Rain rate
moment that is close to the central moments. But, the estimation using the closed-form expression is
problem is the assumed exponential distribution would compared with in situ observation and fixed-power
not be able to provide moments other than the central law-based estimates are presented in Section 4.
moments, such as reflectivity (Z), linear reflectivity Specific propagation phase is derived using Z and ZDR
difference (ZDP), and specific differential phase (KDP). variables. The estimated propagation phase using Z
Thus, an accurate mathematical description of DSD is and ZDR measurements is compared with the actual
essential for estimating rain as well as multi- measured propagation phase for verifying self-
parameter radar observations such as Z, ZDR, and KDP. consistency between Z, ZDR and KDP and also the
applicability of analytically derived relations using a
Ulbrich (1983) suggested the use of the Gamma Λ-µ relation. A summary of the results are given in
distribution for representing a rain drop spectrum as section 5.
n( D ) = N 0 D µ exp( − ΛD ) . (1) 2. DSD parameters using un-truncated and truncated
moment methods
The Gamma DSD with three parameters (N0, µ, and Raindrop size distribution can be measured using
Λ) is capable of describing a broader variation in various instruments such as a momentum impact
raindrop size distribution than an exponential disdrometer, wind profiler, particle measuring probe
distribution which is a special case of Gamma (PMS) and video disdrometer [Yuter, 1997; Williams,2000]. In this study, the video disdrometer
measurements collected in PRECIP’98 are analyzed. ∞
The video disdrometer was operated by Iowa State < D n >= ∫ D n n( D )dD = N 0 Λ − ( µ + n +1)
0 . (2)
University in east-central Florida during the summer
of 1998 when NCAR's S-Pol radar was also deployed Γ( µ + n + 1)
to evaluate the potential of polarimetric radar for
estimating rain in a tropical environment. Following is In general, the three parameters (N0, µ, and Λ) can be
a brief review of the method of fitting the measured solved from any three moments such as the 2nd, 4th
DSDs to Gamma distribution and finding the relations and 6th moment. To eliminate Λ and find µ, a ratio is
among the DSD parameters. defined as
The moment method has been widely accepted in the < D4 >2 ( µ + 3)( µ + 4)
meteorology community because of its robustness in η= = . (3)
obtaining rain rate [Kozu and Nakamura, 1991; Tokay < D >< D >
2 6 ( µ + 5)( µ + 6)
and Short, 1996; Ulbrich, 1998]. In the past, the
integration of most moment calculations is usually Then, µ can be easily solved from (3) as
performed from zero to infinite size range as
__________________________________________
(7 − 11η ) − [(7 − 11η )2 − 4(η − 1)(30η − 12)
µ= ; (4)
2(η − 1)
Λ can be calculated from
1/ 2 1/ 2
< D > Γ( µ + 5) < D > ( µ + 4)( µ + 3)
2 2
Λ = = . (5)
< D > Γ( µ + 3)
4 4
N0 can be calculated from any of the three moments
for specified µ and Λ. to observe a raindrop larger than 8 mm). The typical
range of raindrop size estimated by the Joss
It should be noted that the integration in (2) is disdrometer is between 0.3 mm and 5 mm while a
performed from 0 to infinite; i.e. un-truncated size video disdrometer can measure raindrop size between
distribution. Raindrop distribution is measured over a 0.1 and 8mm. However, the above-described method
finite sample volume and time, hence only a finite for estimating DSD parameters is applicable only for
number of raindrops were counted within a finite size un-truncated DSD. For a Gamma distribution with
range [Dmin, Dmax] because of practical and sampling truncated size range, the statistical moments are
limitation in measuring small and large drops (it is rare calculated as
Dmax
< D n >= ∫ D n n( D)dD = N 0Λ −( µ + n+1) [γ ( µ + n + 1, ΛDmax ) − γ ( µ + n + 1, ΛDmin )] (6)
Dmin
An accurate way of estimating DSD parameters for a
truncated spectrum should be based on the truncated
where γ(…) is an incomplete Gamma function. As moments as shown in Eq. (6) and to calculate the
expected, the truncated moments depend on the upper integral parameters used in DSD parameter retrieval
and lower limit of droplet size in the measured consistent with the truncated (or measured) raindrop
spectrum. If the moments obtained from (6) are used size range.
to fit Gamma distribution shown in Eq. (1) using the
above un-truncated moment method described in Using the expressions of truncated moments, Eqs. (3)
Eq.(2-5), the resultant DSD parameters may be in and (5) become
error.[γ ( µ + 5, Λ Dmax ) − γ ( µ + 5, Λ Dmin )]
2
η= (7)
γ ( µ + 3, Λ D max )γ ( µ + 7, Λ Dmax ) − γ ( µ + 3, Λ Dmin )γ ( µ + 7, Λ Dmin )
and
1/ 2
< D 2 > [γ ( µ + 5, ΛDmax ) − γ ( µ + 5, ΛDmin )]
Λ = (8)
< D 4 > [γ ( µ + 3, ΛDmax ) − γ ( µ + 3, ΛDmin )]
Eqs. (7) and (8) constitute joint equations for µ and Λ µ = −0.036Λ2 + 1.2385Λ − 2.286 (9)
for the truncated moments that are difficult to separate
from each other. Since the above equations are non-
or
linear, an iterative approach is used for solving µ and
Λ.
Λ = 0.037 µ 2 + 0.691µ + 1.926 (10)
2.2 Analysis of µ - Λ relation
without truncation, and
Video disdrometer measurements collected in
PRECIP’98 are used in this study. The data set is the
µ = −0.040Λ2 + 1.405Λ − 2.461 (11)
same as that reported in Zhang et al., 2001, except
minor revisions were made for splashing and wind
effects during DSD measurements. The video or
disdrometer was operated by Iowa State University in
east-central Florida during the summer. We use the Λ = 0.0365µ 2 + 0.735µ + 1.935 (12)
above moment and truncated moment methods to fit
the measured DSDs with Gamma distribution. The for the estimate using the truncated moment method.
Gamma DSD parameters are calculated and analyzed.
Figure 1 shows the scatter plots of the fitted DSD It is interesting to note that the µ and Λ relations do
parameters (µ vs Λ). Figure 1a is obtained from the not change much while the mean values of µ and Λ
un-truncated moment method and Figure 1b from the change from 4.09, 5.58 in Figure 1c to 3.25, 4.92 in
truncated moment method. There are a total of 1341 Figure 1d for the truncated moment method.
data points covering 22 hrs and 21 minutes in 17 days
Theoretically either µ(Λ) or Λ(µ) can be used. In
during PRECIP’98. Both Figures 1a and 1b show
practice, however, there are some differences. We
correlation between µ and Λ. However, retrievals of µ
notice that µ(Λ) fits large values better while Λ(µ) is a
and Λ obtained using the truncated moment method better fit for small values. Since heavy rain tends to
show better correlation than the corresponding set
have small values of µ and Λ, we expect that Λ(µ) is a
retrieved using the untruncated moment method.
better approach. To quantify the difference, the
Further analysis of raindrop spectra revealed the
correlation coefficients between the fitted value and
correlation between µ and Λ is also reduced due to
true value of µ and Λ were compared. They both have
incomplete sampling of DSD as a result of finite
a correlation coefficient of 0.973. However, when µ
sampling volume of the video-disdrometer within a 1
minute sample time. To minimize the error due to and Λ are weighted by corresponding rain rate, the
sampling effects, data were filtered by allowing only correlation coefficient increases to 0.956 for µ and
those with large rain rate >5 mm hr-1 and number of 0.982 for Λ. This supports the fact that the Λ(µ)
raindrops > 1000. The revised plot with the above- relation is better than µ(Λ) for high rain rate cases.
mentioned threshold are shown in Figures 1c and 1d.
The figures contain only 248 data points but captured 3. Relations among statistical moments
75% of the rainfall amount in Figures 1a,b. The
scatter plots shown in Figure 1c and 1d show less As mentioned in the Introduction, the three parameters
scatter and correlation between µ - Λ is higher. A of Gamma DSD are difficult to retrieve from limited
relation between Λ and µ is estimated using a radar measurements. The relation Λ(µ) or µ(Λ)
polynomial least-squares fit, and is given as derived in the previous section constitutes a
constrained condition for Gamma distribution. The µ-
Λ relation applied to Gamma DSD shown in Eq. (1)Figure 1: Scatter plots of µ-Λ values obtained using moment method and truncated moment method. The plots in first
row include all the DSDs collected during 22 hrs and 21 minutes in 17 day of observations. The plots in second row
include measured DSDs only for R > 5 mm hr-1 and total raindrop counts > 1000.
reduces to a two parameter DSD and is dubbed as a constrained Gamma DSD can be found by taking a
constrained Gamma DSD. ratio as
Statistical moments are integral parameters of a DSD. < Dk > Γ( µ + k + 1)
They are directly related to polarimetric radar = Λ ( µ ) − ( k −l ) , (13)
measurements and rainfall rate. For a radar wavelength l Γ( µ + l + 1)
large compared to hydrometeors as in the case of S-
band radar measurement of rain: the reflectivity factor i.e.
is the 6th moment (Z = ); specific differential
phase is proportional to the 4.6th moment (KDP ∝ Γ( µ + k + 1)
) and rain rate is related to the 3.67th moment (R < D k >= Λ ( µ ) − ( k −l ) < Dl > . (14)
Γ( µ + l + 1)
~ ). It is important to know the relation among
the moments and hence the relations between radar
measurements and rain parameters. Instead of It should be pointed out that Eq. (14) does not specify
eliminating the median volume diameter as in Ulbrich, a linear relation between the moments. Actually it
1983; Testud, et al.; 2001, we find the relation does not even guarantee a functional relation since all
between two moments by canceling N0. Therefore, a of the moments depend on µ which is a variable. A
relation between the kth and lth moments of a linear relation exists only when the shape of DSD is
fixed such as the equilibrium shape of DSD [List, et
al., 1988] or constrained Gamma DSD with a constantµ in (14). The shape of DSD (or µ) however, usually ranges of rain rate than typical scatter in Z-R relations
depends on rain type and rain rate, and should not be [Doviak and Zrnic, 1993; Ulbrich and Atlas, 1998].
treated as a constant. For rain estimation from specific differential phase, a
relation of R = 40.56 K DP is currently used. It is
0.866
The relation between moments can be used for noted that the relation tends to underestimate rain,
specifying rain rate as a function of various radar especially for low rainfall [Brandes et al., 2001].
measurements. Since the reflectivity factor is the 6th Recently, ZDR was combined with KDP to reduce
moment, rain rate (R in mm hr-1 ) is proportional to the effects due to an apparent change in drop shape as a
3.67th moment given as result of canting and oscillation [Ryzhkov and Zrnic,
1995]. Similar to the procedure for deriving the R(Z,
R = 7.125 × 10 −3 < D 3.67 > . (15) ZDR) relation, a physically-based R(KDP,ZDR) relation
can be obtained as follows:
Letting k = 3.67 and l = 6 in Eq. (14) and substituting
into (15), we have 72.28 −(1.118−0.240σ φ +3.495σ φ2 )
R= K DP Z DR . (20)
(1 − 2σ φ2 )
Γ ( µ + 4.67)
R = 7.125 × 10 −3 [ Λ ( µ )] 2.33 Z (16)
Γ ( µ + 7)
The coefficient of 72.28 is almost twice that currently
being used 40.56. Therefore, it is expected Eq. (20)
where will provide a better rain rate estimation, especially for
low rain rate.
Γ( µ + 4.67)
F ( µ ) = [Λ ( µ )]2.33 . (17)
Γ( µ + 7) Specific differential phase KDP is an important
parameter which can be used for a self-consistent
check among Z, ZDR and KDP and attenuation
This shows that the R-Z relation is governed by the correction of reflectivity due to rain. KDP in deg km-1
shape parameter µ. For constrained Gamma DSDs, µ is related to Z and ZDR in linear units as
uniquely determines ZDR [Zhang et al. 2001] for
assumed axis ratio and canting angle of raindrops.
Thus F(µ) as a function of ZDR can be obtained as K DP = 5.304 × 10−5 (1 − 2σ φ2 )
. (21)
− (2.011− 0.423σ φ + 6.27σ φ2 )
−bR ZZ
R = 3.73 × 10 −3 Z Z DR (18)
DR
In the following section the above-derived closed
where the exponent bR depends on the effective expressions are used for estimating rain rate and also
canting angle of raindrops. Assuming the mean verifying self-consistency among Z, ZDR and KDP.
canting angle is zero, the coefficient bR is expressed as
a function of the standard derivation of effective 4. Rain Rate Estimation
canting angle (σφ ) as
During PRECIP’98, NCAR's S-Pol radar was
b R = 3.130 − 0.667σ φ + 9.77σ φ . 2
(19) deployed for evaluating the potential of polarimetric
radar for estimating rain in a tropical environment.
The rain gauge measurements were also collected
where σφ is in radian. Compared with the widely used during the project. A detailed description of the project
−3 −4.86
relation R = 4.84 × 10 Z Z DR [Sachidananda and instrument deployment is illustrated in Brandes, et
and Zrnic, 1987], Eq. (18) has a smaller coefficient al., 2001.
and lower value of exponent for ZDR. In general, it
gives a lower rain rate estimation, which agrees better 4.1 Comparisons of rain estimation
with gauge measurements [Brandes et al., 2001] for
specified Z and ZDR. Figure 2 shows the relation Figure 3 shows comparisons between radar estimated
between Z and R for various ZDR. As we expect, a rain rate and the corresponding rain gauge
fixed Z gives a higher rain rate for a smaller ZDR than measurements. The rain rate is plotted as a function of
with a larger ZDR because small ZDR is related to a time for the rain gauge location.
larger amount of smaller raindrops. Specification of The radar estimated rain is calculated using both the
ZDR confines the Z-R relation constrained to smaller classic relations and the newly derived relations in this
paper for comparison. For the result using classic
relations (Figure 3a), both R(Z) and R(KDP)Figure 2: Relationship between reflectivity and rain rate for various differential reflectivity factors. Larger ZDR
corresponds small µ. The thickness of each line represents the margin due to effective canting angle of raindrops.
underestimate rain by 40% while R(Z, ZDR) 4.3 Indirect verification of µ-Λ relation and self-
overestimate rain by more than 20%. This shows the consistency in polarization radar observation
inconsistency among the empirical relations. The
inconsistency has also been found in large data sets To verify the validity of constrained Gamma rain DSD
[Brandes et al., 2001] and self-consistency in polarization observation, we
compare the estimated differential phase and specific
Using the relations derived from constrained Gamma differential phase with those measured. The estimated
DSD, we plot the estimated rain rate in Figure 3b. The KDP is obtained from power measurements; i.e., Z and
physically-based relations are shown to give consistent ZDR using Eq. (21) with a specified σφ = 12o
results and agree with the gauge measurement much [Ryzhkov et al., 1999] and the estimated φDP is
better than the results obtained using classic relations calculated as φ DP
e
= 2∫ K DP
e
(l ) dl where l is the
based on a fixed power law. This is because the
distance along the radial. Figure 4 shows comparisons
constrained Gamma DSD based relations use the DSD
of differential phase between measurements and
information directly. The classic relations, however,
estimations. Figure 4a is an example of differential
were obtained using least-squares fitting, which
depends on the selection of DSD data sets that were phase plotted as a function of range. As expected,
chosen for the fitting; for example, DSDs with a estimated φDP monotonically increases with range and
certain range of DSD parameters and weight might agrees well with the mean of the measurement. The
not be a true representation of natural rain drop size statistical comparison is shown in the scatter plot in
distribution. Figure 4b. There are 20 rays and 1966 data points. The
mean of the measured φDP is 11.65 degrees and the
mean of the estimated value is 11.77 degrees. The bias
is negligible.(a)
(b)
Figure 3: Comparison between various rain rate estimators and gauge measurements. (a) Fixed power-law rain
estimators are used for rain retrieval. Polarization radar-based rain accumulation show both under and over
estimation compared to the gauge observations and (b) Analytically derived rain estimators produce almost same
rain accumulation.
5. Summary and Discussions DSD allows the DSD parameters retrieved from
polarization radar measurements: reflectivity and
Analytical relations for rain rate estimation and self- differential reflectivity. General relations among the
consistency among Z, ZDR and KDP were investigated moments and physical quantities such as reflectivity,
using constrained Gamma rain DSD in this paper. The propagation phase and rain rate were derived. The
constrained condition (µ - Λ relation) is derived from physically-based relations are used for a rain
video-disdrometer measurements. The two parameter estimation and self-consistency check. It has been(a)
(b)
Figure 4: Indirect verification of Λ-µ relation and self-consistency among polarization radar observations. (a) An
example of measured and estimated φDP along a radar beam, and (b) scatter plot of measured and estimated φDP a
number of segments.
shown that: (1) the µ - Λ relation holds at least for the measurement with results being consistent with in situ
data collected in sub-tropical rain events such as in observations.
Florida, and (2) physically-based relations show
smaller bias in rain estimation from polarimetric radar Polarimetric measurements are sensitive to DSD,
shape and canting angle. Even though the µ - Λrelation simplifies DSD representation, any difference Haddad, Z.S., S.L. Durden and E. Im, 1996:
between assumed and actual microphysical parameters Parameterizing the raindrop size distribution,
such as shape and canting angle might introduce J. Appl. Meteor, 35, 3-13.
significant uncertainties in polarization radar-based Illingworth, A.J. and T.M. Blackman, 1999:The need
retrieval. The equilibrium shape of raindrops is to normalize RSD based on the Gamma RSD
assumed in this study while some observations suggest formulation and implication for interpreting
that a more spherical shape should be adapted. The polarimetric radar data. Preprints, 29th Int.
equilibrium shape of raindrops is assumed for Conf. On Radar Meteor., Montreal. Amer.
maintaining continuity with earlier studies. A more Meteor. Soc., Boston, 629-631.
accurate model may be used in future studies when the Jameson, A.R., 1983a: Microphysical interpretation
axis ratio and canting angle are well understood. of multiparameter radar measurements in rain
– Part I: Interpretation of polarization
Acknowledgements measurements and estimation of raindrop
shapes, J. Atmos. Sci., 40,1792-1802.
The authors wish to thank Witold F. Krajewski and Jameson, A.R., 1983b: Microphysical interpretation
Anton Kruger of Iowa State University for making the of multiparameter radar measurements in rain
video disdrometer data available. They would also like – Part I: Estimation of raindrop distribution
to thank J. Lutz and M. Randall of NCAR/ATD for parameters by combined dual-wavelength and
smooth operation of S-Pol radar. The study was partly polarization measurements, J. Atmos. Sci.,
supported by funds from the National Science 40,1803-1813.
Foundation that have been designated for the U.S. Kozu, T., and K. Nakamura, 1991: Rain parameter
Weather Research Program at National Center for estimation from dual-radar measurements
Atmospheric Research (NCAR). combining reflectivity profile and path-
integrated attenuation, J. Atmos. and Ocean.
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